Brauer Groups, Tamagawa Measures, and Rational Points on Algebraic Varieties

Home , Gaussian rational

Mathematical Surveys and Monographs Volume 198

Jörg Jahnel

American Mathematical Society

Brauer Groups, Tamagawa Measures, and Rational Points on Algebraic Varieties

Mathematical Surveys and Monographs Volume 198

Brauer Groups, Tamagawa Measures, and Rational Points on Algebraic Varieties

Jörg Jahnel

American Mathematical Society Providence, Rhode Island

EDITORIAL COMMITTEE Ralph L. Cohen, Chair Benjamin Sudakov Robert Guralnick MichaelI.Weinstein MichaelA.Singer

2010 Mathematics Subject Classiﬁcation. Primary 11G35, 14F22, 16K50, 11-04, 14G25, 11G50.

For additional information and updates on this book, visit www.ams.org/bookpages/surv-198

Library of Congress Cataloging-in-Publication Data Jahnel, J¨org, 1968– Brauer groups, Tamagawa measures, and rational points on algebraic varieties / J¨org Jahnel. pages cm. — (Mathematical surveys and monographs ; volume 198) Includes bibliographical references and index. ISBN 978-1-4704-1882-3 (alk. paper) 1. Algebraic varieties. 2. Geometry, Algebraic. 3. Brauer groups. 4. Rational points (Ge- ometry) I. Title.

QA564.J325 2014 516.353—dc23 2014024341

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Contents

Preface vii Introduction 1 Notation and conventions 11

Part A. Heights 13 Chapter I. The concept of a height 15

1. The naive height on the projective space over É 15 2. Generalization to number ﬁelds 17 3. Geometric interpretation 21 4. The adelic Picard group 25 Chapter II. Conjectures on the asymptotics of points of bounded height 35 1. A heuristic 35 2. The conjecture of Lang 38 3. The conjecture of Batyrev and Manin 40 4. The conjecture of Manin 44 5. Peyre’s constant I—the factor α 47 6. Peyre’s constant II—other factors 50 7. Peyre’s constant III—the actual deﬁnition 59 8. The conjecture of Manin and Peyre—proven cases 62

Part B. The Brauer group 81 Chapter III. On the Brauer group of a scheme 83 1. Central simple algebras and the Brauer group of a ﬁeld 84 2. Azumaya algebras 89 3. The Brauer group 93 4. The cohomological Brauer group 94 5. The relation to the Brauer group of the function ﬁeld 98 6. The Brauer group and the cohomological Brauer group 101 7. The theorem of Auslander and Goldman 103 8. Examples 107 Chapter IV. An application: the Brauer–Manin obstruction 119 1. Adelic points 119 2. The Brauer–Manin obstruction 122 3. Technical lemmata 126 4. Computing the Brauer–Manin obstruction—the general strategy 129 5. The examples of Mordell 132

vi contents

6. The “ﬁrst case” of diagonal cubic surfaces 146 7. Concluding remark 161

Part C. Numerical experiments 163

Chapter V. The Diophantine equation x4 +2y4 = z4 +4w4 165 Numerical experiments and the Manin conjecture 165 1. Introduction 166 2. Congruences 167 3. Naive methods 169 4. An algorithm to eﬃciently search for solutions 169 5. General formulation of the method 171 6. Improvements I—more congruences 172 7. Improvements II—adaptation to our hardware 176 8. The solution found 182 Chapter VI. Points of bounded height on cubic and quartic threefolds 185 1. Introduction—Manin’s conjecture 185 2. Computing the Tamagawa number 189 3. On the geometry of diagonal cubic threefolds 193 4. Accumulating subvarieties 195 5. Results 199 Chapter VII. On the smallest point on a diagonal cubic surface 205 1. Introduction 205 2. Peyre’s constant 208 3. The factors α and β 209 4. A technical lemma 211 5. Splitting the Picard group 212 6. The computation of the L-function at 1 216 7. Computing the Tamagawa numbers 219 8. Searching for the smallest solution 221 9. The fundamental ﬁniteness property 222 10. A negative result 233 Appendix 239 1. A script in GAP 239 2. The list 241 Bibliography 247 Index 261

Preface

In this book, we study existence and asymptotics of rational points on algebraic varieties of Fano and intermediate type. The book consists of three parts. In the ﬁrst part, we discuss to some extent the concept of a height and formulate Manin’s conjecture on the asymptotics of rational points on Fano varieties. In the second part, we study the various versions of the Brauer group. We explain why a Brauer class may serve as an obstruction to weak approximation or even to the Hasse principle. This includes two sections devoted to explicit computations of the Brauer–Manin obstruction for particular types of cubic surfaces. The ﬁnal part describes numerical experiments related to the Manin conjecture that were carried out by the author together with Andreas-Stephan Elsenhans.

Prerequisites. We assume that the reader is familiar with some basic mathematics, including measure theory and the content of a standard course in algebra. In addition, a certain background in some more advanced fields shall be necessary. This essentially concerns three areas. a) We will make use of standard results from algebraic number theory and class field theory, as well as such concerning the cohomology of groups. The content of articles [Cas67, Se67, Ta67, A/W, Gru] in the famous collection edited by J. W. S. Cassels and A. Fröhlich shall be more than sufficient. Here, the most important results that we shall use are the existence of the global Artin map and, related to this, the computation of the Brauer group of a number field [Ta67, 11.2]. b) We will use the language of modern algebraic geometry as described in the textbook of R. Hartshorne [Ha77, Chapter II]. Cohomology of coherent sheaves [Ha77, Chapter III] will be used occasionally. c) In Chapter III, we will make substantial use of étale cohomology. This is probably the deepest prerequisite that we expect from the reader. For this reason, we will formulate its main principles, as they appear to be of importance for us, at the beginning of the chapter. It seems to us that, in order to follow the arguments, an understanding of Chapters II and III of J. Milne’s textbook [Mi] should be sufficient. At a few points, some other background material may be helpful. This concerns, for example, Artin L-functions. Here, [Hei] may serve as a general reference. Pre- cise citations shall, of course, be given wherever the necessity occurs.

A suggestion that might be helpful for the reader. Part C of this book describes experiments concerning the Manin conjecture. Clearly, the particular samples are

vii

viii preface chosen in such a way that not all the diﬃculties, which are theoretically possible, really occur. It therefore seems that Part C might be easier to read than the others, particularly for those readers who are very familiar with computers and the concept of an algorithm. Thus, such a reader could try to start with Part C to learn about the experiments and to get acquainted with the theory. It is possible then to continue, in a second step, with Parts A and B in order to get used to the theory in its full strength.

References and citations. When we refer to a definition, proposition, theorem, etc., in the same chapter we simply rely on the corresponding numbering within the chapter. Otherwise, we add the number of the chapter. For the purpose of citation, the articles and books being used are encoded in the manner specified by the bibliography. In addition, we mostly give the number of the relevant section and subsection or the number of the definition, proposition, theorem, etc. Normally, we do not mention page numbers.

Acknowledgments. I wish to acknowledge with gratitude my debt to Y. Tschinkel. Most of the work described in this book, which is a shortened version of my Habil- itation Thesis, was initiated by his numerous mathematical questions. During the years he spent at Göttingen, he always shared his ideas in an extraordinarily gen- erous manner. I further wish to express my deep gratitude to my friend and colleague Andreas- Stephan Elsenhans. He inﬂuenced this book in many ways, directly and indirectly. It is no exaggeration to say that most of what I know about computer programming I learned from him. The experiments, which are described in Part C, were carried out together with him as a joint work. I am also indebted to Stephan for proofreading. The computer part of the work described in this book was executed on the Sun Fire V20z Servers of the Gauß Laboratory for Scientiﬁc Computing at the Göttingen Mathematical Institute.The author is grateful to Y. Tschinkel for permission to use these machines as well as to the system administrators for their support.

Jörg Jahnel Siegen, Germany Spring 2014

Introduction

Here, in the midst of this sad and barren landscape of the Greek ac- complishments in arithmetic, suddenly springs up a man with youthful energy: Diophantus. Where does he come from, where does he go to? Who were his predecessors, who his successors? We do not know. It is all one big riddle. He lived in Alexandria. If a conjecture were permitted, I would say he was not Greek; ... if his writings were not in Greek, no-one would ever think that they were an outgrowth of Greek culture ... . Hermann Hankel (1874, translated by N. Schappacher)

Diophantine equations have a long history. More than two thousand years ago, Diophantus of Alexandria considered, among many others, the equations

x2 + y2 = z2 , (∗)

y(6 − y)=x3 − x, and y2 = x2 + x4 + x8 .

In Diophantus’ book Arithmetica, we ﬁnd the formula (p2 − q2)λ, 2pqλ, (p2 + q2)λ (†) that generates inﬁnitely many solutions of (∗). For the second and third of the equations mentioned, Diophantus gives particular solutions, namely (1/36, 1/216) and (1/2, 9/16), respectively. In general, a polynomial equation in several indeterminates, where solutions are sought in integers or rational numbers, is called a Diophantine equation in honour of Diophantus. Diophantus himself was interested in solutions in positive integers or positive rational numbers. Contrary to the point of view usually adopted today, he did not accept negative numbers.

It is remarkable that algebro-geometric methods have often been fruitful in order to understand a Diophantine equation. For example, there is a simple geometric idea behind formula (†). Indeed, since the equation is homogeneous, it suﬃces to look for solutions 2 2 of X + Y =1in rationals. This equation deﬁnes the unit circle. For every t ∈ Ê, there is the line “x = −ty +1” going through the point (1, 0). An easy calculation

2 introduction shows that the second point where this line meets the unit circle is given by 1 − t2 2t , . (‡) 1+t2 1+t2 As every point on the unit circle may be connected with (1, 0) by a line, one sees that the parametrization (‡) yields every rational point on the circle (except for (−1, 0), for which the form of line equation given is not adequate). Consequently, formula (†) delivers essentially every solution of equation (∗), a fact which was seemingly not known to the ancient mathematicians. The morphism

1 −→ 2 P P , (p : q) → (p2 − q2):2pq :(p2 + q2) provides a rational parametrization of the plane conic C given by the equation x2 +y2 = z2 in P2. More or less the same method works for every conic in the plane. Further, it may be extended to several classes of singular curves of higher degree.

Every Diophantine equation deﬁnes an algebraic variety X in an aﬃne or projective space. There is a one-to-one correspondence between solutions of the Diophan-

tine equation and É-rational points on X. We will prefer geometric language to number theoretic throughout this book. The cases in which there is an obvious rational parametrization are, in some sense, the best possible. But even when there is nothing like that, algebraic geometry often yields a guideline of which behaviour to expect—whether there will be no, a few, or many solutions. The Kodaira classification distinguishes between Fano varieties, varieties of intermediate type, and varieties of general type (at least under the additional assumption that X is non-singular). It does not use any specifically arithmetic information, but only information about X as a complex variety. Nevertheless, there is overwhelm- ing evidence for a strong connection between the classification of X according to Kodaira and its set of rational points. To make a vague statement, on a Fano variety, there are infinitely many rational points expected while, on a variety of general type, there are only finitely many rational points or even none at all. More precisely, there is the conjecture that, on a Fano variety, there are always infinitely many rational points after a suitable finite extension of the ground field. On the other hand, for varieties of general type, there is the conjecture of Lang. It states that there are only finitely many rational points outside the union of all closed subvarieties that are not of general type.

Another method to analyze a Diophantine equation is given by congruences. Kurt Hensel provided a more formal framework for this method by his invention of the p-adic numbers. As one is working over local ﬁelds, this might be called the local method. Consider, for example, the Diophantine equation

x3 +7y3 +49z3 +2u3 +14v3 +98w3 =0. (§)

introduction 3

It has no solution in É except for (0, 0, 0, 0, 0, 0). This may be seen by a repeated application of an argument modulo 7. A more formal reason is the fact that the

projective algebraic variety deﬁned by (§) has no point deﬁned over É7.

One might ask whether or to what extent solvability over Ép for every prime number p together with solvability in real numbers implies the existence of a rational solution. This question has been very inspiring for research over many decades. As early as 1785, A.-M. Legendre gave an aﬃrmative answer for equations of the type q(x, y, z)=0, where q is a ternary quadratic form. Legendre’s result was generalized to quadratic forms in arbitrarily many variables by H. Hasse and H. Minkowski. The term “Hasse principle” was coined to describe the phenomenon. A totally diﬀerent sort of examples where the Hasse principle is valid is provided by the circle method originally developed by G. H. Hardy and J. E. Littlewood. The circle method uses tools from complex analysis to study the asymptotics of the number of points of bounded height on complete intersections in a very high- dimensional projective space. It provides an asymptotic formula and an error term. The main term is of the form

− − − τBn+1 d1 ... dr

n for a complete intersection of multidegree (d1, ... ,dr) in P . The reader might want to consult [Va] for a description of the method and references to the original literature. The exponent of the main term allows a beautiful algebro-geometric interpretation. The anticanonical sheaf on a complete intersection of multidegree (d1, ... ,dr) n in P is precisely O(n +1− d1 − ... − dr)|X . This means, when working with an anticanonical height instead of the naive height, the circle method proves linear

growth for the É-rational points. The coeﬃcient τ of the main term is a product of p-adic densities together with a factor corresponding to the Archimedean valuation. Unfortunately, it is necessary to make very restrictive assumptions on the number of variables in comparison with the degrees of the equations. These assumptions on the dimension of the ambient projective space are needed in order to ensure that the provable error term is smaller than the main term. One might, nevertheless, hope that there is a similar asymptotic under much less restrictive conditions. This is the origin of Manin’s conjecture. However, as was observed by J. Franke, Yu. I. Manin, and Y. Tschinkel [F/M/T], the main term as described above is not compatible with the formation of direct products. Already on a variety as simple as P1 ×P1, the growth of the number

of the É-rational points is actually asymptotically equal to τBlog B.Thismaybe seen by a calculation, which is completely elementary. Thus, in general, the asymptotic formula has to be modiﬁed by a log-factor. − Franke, Manin, and Tschinkel suggest the factor logrk Pic(X) 1B and prove that this factor makes the asymptotic formula compatible with direct products.

4 introduction

Furthermore, it turns out that the coefficient τ has to be modified when rk Pic(X) > 1. There appears an additional factor, which today is called α(X). This factor is defined by a beautiful yet somewhat mysterious elementary geometric construction.

Another problem is that the Hasse principle does not hold universally. Con- sider the following elementary example, which was given by C.-E. Lind in 1940. Lind [Lin] dealt with the Diophantine equation

2u2 = v4 − 17w4 deﬁning an algebraic curve of genus 1. It is obvious that this equation is non- trivially solvable in reals, and it is easy to check that it is non-trivially solvable

in Ép for every prime number p. On the other hand, there is no solution in rationals except for (0, 0, 0). Indeed, assume the contrary. Then there is a solution in integers such that gcd(u, v, w)=1. For such a solution, one clearly has 17 u.Since2 is a square but not a fourth power u − modulo 17, we conclude that 17 = 1. On the other hand, for every odd prime 4 − 4 ≡ 17 divisor p of u, one hasv 17w 0(mod p). This shows p =1.Bythelowof p u quadratic reciprocity, 17 =1. Altogether, 17 =1, which is a contradiction. One might argue that this example is not too interesting since, on a curve of genus 1,

there are relatively few É-rational points to be expected. Thus, it might happen that there are none of them without any particular reason. However, several other counterexamples to the Hasse principle had been invented. Some of them were Fano varieties. For example, Sir Peter Swinnerton-Dyer [SD62] and L.-J. Mordell [Mord] (cf. our Chapter IV, Section 5) constructed examples of cubic surfaces violating the Hasse principle. A few years later, J. W. S. Cas- sels and M.J.T. Guy [Ca/G]aswellasA.Bremner[Bre] even found isolated examples of diagonal cubic surfaces showing that behaviour. Typically, the proofs were a bit less elementary than Lind’s in that sense that they required not the quadratic but the cubic or biquadratic reciprocity low.

In the late 1960s, Yu. I. Manin [Man] made the remarkable discovery that all the known counterexamples to the Hasse principle could be explained in a uniform manner. There was actually a class α ∈ Br(X) in the Brauer group of the underlying

algebraic variety responsible for the lack of É-rational points.

This may be explained as follows. The Brauer group of É is relatively complicated. One has, by virtue of global class ﬁeld theory,

É ⊕ −→ É Br(Spec É)=ker s: / 2 / / .

p prime

Here, s is just the summation. The summand É/ corresponding to the prime Ê number p is nothing but Br(Spec Ép) while the last summand is Br(Spec ). ∈

Let α Br(X) be any Brauer class of a variety X over É. An adelic point

∈ É x =(xν )ν∈Val( É) X( )

introduction 5 É deﬁnes restrictions of α to Br(Spec Ê) and Br(Spec p) for each p. Ifthesumofall

invariants is diﬀerent from zero, then, according to the computation of Br(Spec É),

x may not be approximated by É-rational points. As α ∈ Br(X) then “obstructs” x from being approximated by rational points, the expression Brauer–Manin obstruction became the general standard for this famous observation of Manin. In the counterexamples to the Hasse principle, which were known to Manin in those days, one typically had a Brauer class, the restrictions of which had a totally degenerate behaviour. For example, on Lind’s curve, there is a Brauer class α such

that its restriction is independent of the choice of the adelic point. α restricts to

É É zero in Br(Spec Ê) and Br(Spec p) for p =17but non-trivially to Br(Spec 17).

This suﬃces to show that there is no É-rational point on that curve.

⊆ É In general, the Brauer–Manin obstruction deﬁnes a subset X( É ) X( ) consisting of the adelic points that are not aﬀected by the obstruction. At least for cubic surfaces, there is a conjecture of J.-L. Colliot-Thélène stating that

X( É ) is equal to the set of all adelic points that may actually be approximated

by É-rational points.

∅ ∅ É Thus, X( É ) = , while X( ) = means that X is a proven counterexample

É to the Hasse principle. If X( É ) X( ), then we have a counterexample to weak approximation. If Colliot-Thélène’s conjecture were true, then one could say that all cubic surfaces that are counterexamples to the Hasse principle or to weak approximation are of this form.

The Brauer group of an algebraic variety X over an algebraically non-closed field k admits, according to the Hochschild–Serre spectral sequence, a canonical fil- tration into three terms. The first term is given by the image of Br(Spec k) in Br(X). Second, Br(X)/Br(Spec k) has a subgroup canonically isomor- 1 phic to H Gal(k/k), Pic(Xk) . The remaining subquotient is a subgroup of Gal(k/k) Br(Xk) . It turns out that only the second and third parts are relevant for the Brauer–Manin obstruction. The third one causes the so-called transcendental Brauer–Manin obstruction, which is technically difficult. We will not cover the transcendental Brauer–Manin obstruction in this book. The subquotient 1 H Gal(k/k), Pic(Xk) =0is responsible for what might be called the algebraic Brauer–Manin obstruction. In the cases where the circle method is applicable, the Noether–Lefschetz The- orem shows that Pic(Xk)= with trivial Galois operation. Consequently, 1 H Gal(k/k), Pic(Xk) =0, which is clearly sufficient for the absence of the algebraic Brauer–Manin obstruction. This coincides perfectly well with the observation that the circle method always proves equidistribution. By consequence, in a conjectural generalization of the results proven by the

É circle method, one can work with X( É ) instead of X( ) without making any change in the proven cases. However, in the cases where weak approximation fails, this does not give the correct answer, as was observed by D. R. Heath-Brown [H-B92a] in 1992. On a cubic surface such that

1 H Gal(k/k), Pic(Xk) = /3 and a non-trivial Brauer class excludes two thirds

6 introduction of the adelic points, there are nevertheless as many rational points as naively expected. Even more, E. Peyre and Y. Tschinkel [Pe/T] showed experimentally that

1 if H Gal(k/k), Pic(Xk) = /3 and the Brauer class does not exclude any adelic point, then there are three times more rational points than expected. Correspond- ingly, in E. Peyre’s [Pe95a] deﬁnition of the conjectural constant τ, there appears 1 an additional factor β(X):=#H Gal(k/k), Pic(Xk) .

This book is concerned with Diophantine equations from the theoretical and experimental points of view. It is divided into three parts. The ﬁrst part is devoted to the various concepts of a height. In the ﬁrst chapter, we start with the naive height

for É-rational points on projective space. Then our goal is to deliver some insight into the theories, which provide natural generalizations of this simple concept. The very ﬁrst generalization is the naive height for points in projective space deﬁned

over a finite extension of É. Then, following André Weil, we introduce the concept of a height defined by an ample invertible sheaf. This is a height function, which is defined only up to a bounded summand. To overcome this difficulty, one has to work with arithmetic varieties and metrized

invertible sheaves. Arithmetic varieties are schemes projective over Spec .Actu- ally, this leads to a beautiful geometric interpretation of the naive height. X Indeed, let X be a projective variety over É,andlet be a projective model L X of X over Spec . Fix a hermitian line bundle on . Then, according to the

valuative criterion of properness, every É-rational point x on X extends uniquely

→ X L to a -valued point x: Spec . The height function with respect to is then given by ∗ hL (x):=deg x L .

Here, deg denotes the Arakelov degree of a hermitian line bundle over Spec . It turns out that this coincides exactly with the naive height when one works with X = Pn , L = O(1), and the minimum metric, which is deﬁned by min := min . i=0, ... ,n Xi

In general, hL admits a fundamental finiteness property as soon as L is ample. Chapter II is devoted to some of the most popular conjectures concerning rational points on projective algebraic varieties. We discuss Lang’s conjecture, the conjecture of Batyrev and Manin, and, most notably, Manin’s conjecture about the asymptotics of points of bounded height on Fano varieties (Conjecture II.7.3). A large part of the chapter is concerned with E. Peyre’s Tamagawa type number τ(X), the coefficient expected in the asymptotic formula. We discuss in detail all factors appearing in the definition of τ(X). In particular, we give a number of examples, for which we explicitly compute the factor α(X). We mainly consider smooth cubic surfaces of arithmetic Picard rank two.

Part B is the technical heart of the book. It deals with the concepts of a Brauer group and its applications. The third chapter considers A. Grothendieck’s Brauer group for arbitrary schemes. We recall the concept of a sheaf of Azumaya algebras on a scheme and explain how such a sheaf of algebras gives rise to a class in the

introduction 7

2 étale cohomology group Br (X):=Hét(X, m). This is what is called the cohomological Brauer group. On the other hand, a rather naive generalization of the definition for fields yields the concept of the Brauer group. One has Br(X) ⊆ Br (X). In general, the two are not equal to each other. In Section III.7, we give a proof for the Theorem of Auslander and Goldman stating that Br(X)=Br(X) in the case of a smooth surface. This result was originally shown in [A/G] before the actual invention of schemes. The proof of Auslan- der and Goldman was formulated in the language of Brauer groups for commutative rings. However, all the arguments given carry over immediately to the case of a two-dimensional regular scheme. Although better results are available today, most notably Gabber’s Theorem [dJo2], we feel that the proof of the Theorem of Auslander and Goldman gives a good impression of the methods used to compare Br(X) and Br(X). The chapter is closed by computations of Brauer groups in particular examples. In the case of a variety over an algebraically non-closed field, we study the relation- 1 ship of Br(X) with H Gal(k/k), Pic(Xk) . We prove Manin’s formula expressing the latter cohomology group in terms of the Galois operation on a specific set of divisors. For smooth cubic surfaces, one may work with the classes given by the 27 lines. This leads to the result of Sir Peter Swinnerton-Dyer [SD93] that, for a smooth

1 2

cubic surface, H Gal(k/k), Pic(Xk) is one of the groups 0, /2 , /3 , ( /2 ) ,

2 and ( /3 ) . Swinnerton-Dyer’s proof filled the entire article [SD93]andwaslater modified by P. K. Corn in his thesis [Cor]. We discovered that Swinnerton-Dyer’s result may be obtained in a manner, which is rather brute force, but very simple. The Galois group acting on the 27 lines on a smooth cubic surface is a subgroup of W (E 6). There are only350 conjugacy classes 1 of subgroups of W (E6). We computed H Gal(k/k), Pic(Xk) in each of these cases using GAP. This took 28 seconds of CPU time. As an application of Brauer groups, the third chapter is concerned with the Brauer– Manin obstruction. We recall the notion of an adelic point and define the local and

global evaluation maps. An adelic point x =(xν )ν∈Val( É) is “obstructed” from being approximated by rational points if the global evaluation map ev gives a non-zero value ev(α, x) for a certain Brauer class α ∈ Br(X). We then describe a strategy on how the Brauer–Manin obstruction may be explicitly computed in concrete examples. We carry out this strategy for two special types of cubic surfacess, which, as we think, are representative but particularly interesting.

The ﬁrst type is given as follows. Let p0 ≡ 1(mod 3) be a prime number, and

let K/É be the unique cubic ﬁeld extension contained in the cyclotomic exten-

É ∈ sion É(ζp0 )/ . Fix the explicit generator θ K given by − − i

θ := trÉ (ζ 1) = 2n + ζ (ζp0 )/K p0 p0 ∗ 3 i∈( ) p0 − for n := p0 1 . Then consider the cubic surface X ⊂ P3 ,givenby 6 É 3 (i) (i) 2 x3(a1x0 + d1x3)(a2x0 + d2x3)= x0 + θ x1 +(θ ) x2 . i=1

(i) É Here, a1,a2,d1,d2 ∈ .Theθ denote the three images of θ under Gal(K/ ).

8 introduction

Proposition IV.5.3 provides criteria to verify that such a surface is smooth and has p-adic points for every prime p. More importantly, the Brauer–Manin obstruction can be understood completely explicitly. At least for a generic choice of a1,a2,d1,

1 and d2, one has that H Gal(k/k), Pic(Xk) = /3 . Further, there is a class α ∈ Br(X) with the following property. For an adelic point x =(x ) ,thevalueofev(α, x) depends only on the component x . ν ν νp0 Write x =: (t : t : t : t ). Then one has ev(α, x)=0if and only if νp0 0 1 2 3

a1t0 + d1t3 t3 ∗ ≡ is a cube in p . Note that p0 1(mod 3) implies that only every third element ∗ 0 of p0 is a cube. Observe that the reduction of X modulo p0 is given by

3 x3(a1x0 + d1x3)(a2x0 + d2x3)=x0 .

This means, there are three planes intersecting in a triple line. No Ép0 -rational point may reduce to the triple line. Thus, there are three diﬀerent planes to which a Ép0 -rational point x may reduce. The value of ev(α, x) depends only the plane, to which its component x is mapped under reduction. νp0 For instance (cf. Example IV.5.24), for p0 =19, consider the cubic surface X given by 3 (i) (i) 2 x3(x0 + x3)(12x0 + x3)= x0 + θ x1 +(θ ) x2 . i=1

Then, in 19, the cubic equation

x(1 + x)(12 + x) − 1=0

has the three solutions 12, 15,and17. However, in 19, 13/12 = 9, 16/15 = 15, ∅

and 18/17 = 10, which are three non-cubes. This shows that X( É)= . It is easy

∅

to check that X( É ) = . Therefore, X is an example of a cubic surface violating the Hasse principle. We construct a number of similar examples. For instance, Example IV.5.24 de-

1 scribes a cubic surface X such that H Gal(k/k), Pic(Xk) = /3 , but the generating Brauer class does not exclude a single adelic point. One would expect that X satisﬁes weak approximation. Recall that, in similar examples, E. Peyre and Y. Tschinkel [Pe/T] showed experimentally that there are three times more

É-rational points than expected. The historically ﬁrst cubic surface that could be proven to be a counterexample to the Hasse principle was provided by Sir Peter Swinnerton-Dyer [SD62]. We recover Swinnerton-Dyer’s example (cf. Example IV.5.27) for p0 =7, d1 = d2 =1, a1 =1, and a2 =2. L. J. Mordell [Mord] generalized Swinnerton-Dyer’s work by giving a series of examples for p0 =7and a series of examples for p0 =13. Yu. I. Manin men- tions Mordell’s examples explicitly in his book [Man]. He explains these counterexamples to the Hasse principle by a Brauer class. We generalize Mordell’s examples further to the case that p0 is an arbitrary prime such that p0 ≡ 1(mod 3). We conclude Chapter IV by a section on diagonal cubic surfaces. For these, the Brauer–Manin obstruction was investigated in the monumental work [CT/K/S]

introduction 9 of J.-L. Colliot-Thélène, D. Kanevsky, and J.-J. Sansuc. We present an explicit computation of the Brauer–Manin obstruction under a congruence condition that corresponds more or less to the “ﬁrst case” of [CT/K/S]. Our argument is, however, shorter and simpler than the original one. The point is that we make use of the

1 2 2

fact that H Gal(k/k), Pic(Xk) may be only 0, /2 , /3 , ( /2 ) ,or( /3 ) .

2 Further, the group ( /3 ) appears only once in a very particular case. Thus, in

1 order to prove H Gal(k/k), Pic(Xk) = /3 , it is almost suﬃcient to construct an element of order three.

Part C collects two reports on practical experiments. Chapter V is concerned with the Diophantine equation x4 +2y4 = z4 +4w4 . (¶)

This equation gives an example of a K3 surface X deﬁned over É.Itisanopen

question whether there exists a K3 surface over É that has a ﬁnite non-zero number

of É-rational points. X might be a candidate for a K3 surface with this property. (1:0:1:0) and (1 : 0 : (−1) : 0) are two obvious rational points. Sir Peter Swinnerton-Dyer [Poo/T, Problem/Question 6.c)] had publicly posed the problem to ﬁnd a third rational point on X. But no rational points diﬀerent from the two obvious ones had been found in experiments carried out by several people.

We explain our approach to eﬃciently search for É-rational points on algebraic varieties deﬁned by a decoupled equation. It is based on hashing, a method from

computer science. In the particular case of a surface in P3 , our algorithm is of É complexity essentially O(B2) for a search bound of B. In the ﬁnal implementation, we could work with the search bound B =108.Wedis- covered the following solution of the Diophantine equation (¶):

1 484 8014 +2· 1 203 1204 = 9 050 910 498 475 648 046 899 201, 1 169 4074 +4· 1 157 5204 = 9 050 910 498 475 648 046 899 201.

Up to changes of sign, this is the only non-obvious solution of (¶) we know and the only non-obvious solution of height less than 108 [EJ2, EJ3]. The reader probably thinks that this particular equation is not of fundamental importance, and doing so he or she is definitely right. Let us, however, emphasize that Chapter V discusses an efficient point search algorithm, which works in much more generality. The two final chapters show it at work in experimental investigations related to the Manin conjecture for two important families of Fano varieties. In Chapter VI we describe our investigations regarding the particular families “ax3 = by3 + z3 + v3 + w3”, a, b =1, ... ,100,and“ax4 = by4 + z4 + v4 + w4”, a, b =1, ... ,100, of projective algebraic threefolds. We report numerical evidence for the conjecture of Manin in the refined form due to E. Peyre. Our experiments included searching for points, computing the Tamagawa number, and detecting the accumulating subvarieties. Concerning the programmer’s efforts, detection of accumulating subvarieties was the most difficult part of this project. For example, for one the cubic threefolds, the non-obvious lines in Table 1 have

10 introduction

Table 1. Sporadic lines on cubic threefolds

a b Smallest point Point s.t. x =0 19 18 (1:2:3:-3:-5) (0 : 7 : 1 : -7 : -18) 21 6 (1:2:3:-3:-3) (0 : 9 : 1 : -10 : -15) 22 5 (1:-1: 3: 3: -3) (0 : 27 : -4 : -60 : 49) 45 18 (1:1:3:3:-3) (0 : 3 : -1 : 3 : -8) 73 17 (1 : 5 : -2 : 11 : -15) (0 : 27 : -40 : 85 : -96) been found. These are the only non-obvious lines we know and the only ones containing a point of height less than 5000. We describe all the computations that were done as well some background on the geometry of cubic and quartic threefolds. Observe that the lines on a cubic threefold have a particular rich geometry. They form a smooth surface that is

of general type. Our observation that É-rational lines are rare is therefore in coincidence with Lang’s conjecture. In Chapter VII we return to the more standard case of diagonal cubic surfaces. The experiments are analogous to those described in Chapter VI for diagonal cubic and quartic threefolds. The theory is, however, more complicated. The geometric Picard rank is equal to 7 and, in the generic case, there is a Brauer–Manin obstruction to weak approximation excluding precisely two thirds of the adelic points. The factors α(X) and β(X) appearing in the deﬁnition of Peyre’s constant are not always the same and need to be considered. We demonstrate experimentally the connection of Peyre’s constant with the height m(X) of the smallest rational point. Under the Generalized Riemann Hy- C pothesis, we prove that there is no constant C such that m(X) < τ(X) for every diagonal cubic surface. We also prove that, for diagonal cubic surfaces, the recip- 1 1 rocal τ(X) behaves like a height function, i.e., τ(X) admits a fundamental ﬁnite- ness property.

introduction 11

Notation and conventions

We follow standard notation and conventions from algebra, algebraic number the-

ory, and algebraic geometry. More precisely:

É Ê i) We write Æ, , , ,and for the sets of natural numbers, integers, rational numbers, real numbers, and complex numbers, respectively.

ii) We say that a function f is O(g),forB →∞, if there exists a constant C ∈ Ê such that f(B) ≤ Cg(B),forB suﬃciently large. Instead of f = O(g),wealso write f g or g f. iii) For a group G and elements σ1,...,σn ∈ G, we denote the subgroup generated by σ1,...,σn by σ1,...,σn⊆G.

If G is abelian, then Gn ⊆ G is the subgroup consisting of all elements of torsion dividing n. iv) If a group G operates on a set M,thenM G denotes the invariants. We write M σ instead of M σ. v) All rings are assumed to be associative. vi) If R is a ring, then Rop denotes the opposite ring. I.e., the ring that coincides with R as an abelian group, but in which one has xy = z when one had yx = z in R. vii) For R aringwithunit,R∗ denotes the multiplicative group of invertible elements in R. viii) All homomorphisms between rings with unit are supposed to respect the unit elements. ix) By a field, we always mean a commutative field. I.e., a commutative ring with unit, every non-zero element of which is invertible. If K is a field, then Ksep and K denote a fixed separable closure and a fixed algebraic closure, respectively. x) A ring with unit, every non-zero element of which is invertible, is called a skew field. xi) If R is a commutative ring with unit, then an R-algebra is always understood to be a ring homomorphism j : R → A, the image of which is contained in the center of A.AnR-algebra j : R → A is denoted simply by A when there seems to be no danger of confusion. An R-algebra being a skew field is also called a division algebra. xii) If σ : R → R is an automorphism of R,thenAσ denotes the R-algebra σ j σ σ σ R −→ R −→ A.IfM is an R-module, then we put M := M ⊗R R . M is an Rσ-moduleaswellasanR-module. xiii) All central simple algebras are assumed to be finite dimensional over a base field. xiv) For K a number field, we write OK to denote the ring of integers in K. If ν ∈ Val(K) is a non-Archimedean valuation, then the ν-adic completion of K is O denoted by Kν and its ring of integers by Kν .

In the particular case that K = É, we denote by νp the normalized p-adic valuation corresponding to a prime number p. xv) For R a commutative ring, we denote by Spec R theaﬃneschemeconstituted by its spectrum.

12 introduction xvi) The projective space of relative dimension n over a scheme X will be denoted n by PX . We omit the subscript when there is no danger of confusion. xvii) If X is a scheme over a scheme T and Y is a T -scheme, then we also write XY for the fiber product X ×T Y .IfY = Spec R is affine, then we write XR instead of XSpec R. xviii) For X aschemeoveraschemeT , we denote by Xt the fiber of X over t ∈ T . C O If is a scheme over the integer ring Kν of the completion Kν of the number field K with respect to the valuation ν,thenwewriteCν for the special fiber. In the C C particular case that K = É and ν = νp,wewrite p instead of νp . If C is a scheme over the integer ring O of a number field K,thenweusethesame O notation, not mentioning the base change to Kν . xix) For R any commutative ring, A a commutative R-algebra, and X an R-scheme, a morphism x: Spec A → X of R-schemes is also called an A-valued point on X. If A is a field, then we also adopt more conventional language and speak of a point defined or rational over A.ThesetofallA-valued points on X will be denoted by X(A). xx) If C is a scheme over a valuation ring O and x ∈ C (O), then the reduction of x is denoted by x.

Part A

Heights

CHAPTER I

The concept of a height

Equations are just the boring part of mathematics. I attempt to see things in terms of geometry. Stephen Hawking (A Biography (2005) by Kristine Larsen, p. 43)

1. The naive height on the projective space over É

1.1. Heights have been studied by number theorists for a very long time. A height is a function measuring the size or, more precisely, the arithmetic complexity of certain objects. These objects are classically solutions of Diophantine equations or rational points on an algebraic variety. A height then might answer the question, How many bits would one need in order to store the solution or the point on a computer? More recently, starting with G. Faltings’ ideas of heights on moduli spaces, it became more common to consider heights for more complicated objects, such as cycles.

n 1.2. Deﬁnition. For (x0 : ... : xn) ∈ P ( É), put

| | Hnaive(x0 : ... : xn):= max xi . i=0,... ,n Here (x0 : ... : xn)=(x0 : ... : xn),

such that all xi are integers and gcd(x0,...,xn)=1.

n → Ê The function H : P ( É) is called the naive height.

naive É

1.3. Fact (Fundamental ﬁniteness). For every B ∈ Ê, there are only ∈ n

ﬁnitely many points x P ( É) such that É

Hnaive(x)

− Proof. For each component of (x0, ... ,xn),wehave B

1.4. The naive height is probably the simplest function one might think of that fulfills the fundamental finiteness property. For more general height functions, the fundamental finiteness property will always be required.

16 the concept of a height [Chap. I

1.5. Remark. Let X ⊂ P be a subvariety. Then every É-rational point É

n É on X is also a É-rational point on P . We call the restriction of Hnaive to X( ) the naive height on X. It is obvious that the naive height on X fulﬁlls the fundamental ﬁniteness property.

1.6. Notation. i) For a prime number p, denote by . p the normalized p- vp ∈ \{ } ± v2 · v3 · · k adic valuation. I.e., for x É 0 ,letx = 2 3 ... pk be its decomposition into prime factors. Then put −vp x p := p .

Further, 0 p := 0. ii) We use . ∞ as an alternative notation for the usual absolute value, x if x ≥ 0 , x ∞ := −x if x<0 .

1.7. Fact. For p a prime number or inﬁnity, . p is indeed a valuation. ∈ This means, for x, y É, i) x p ≥ 0, ii) x p =0if and only if x =0, iii) xy p = x p · y p, iv) x + y p ≤ x p + y p.

For p = ∞, one even has that x + y p ≤ max{ x p, y p}.

\{ } 1.8. Fact (Product formula). For x ∈ É 0 , one has x p =1. p prime or ∞

1.9. Lemma. Let (x : ... : x ) ∈ P ( É).Then

0 n É Hnaive(x0 : ... : xn)= max xi p . i=0,... ,n p prime or ∞ Proof. The product formula implies that the right-hand side remains unchanged when (x0 : ... : xn) is replaced by (λx0 : ... : λxn) for λ =0 .Thus,wemay suppose that all xi are integers and gcd(x0,...,xn)=1. These assumptions imply that

max xi p =1 i=0,... ,n for every prime number p. Hence, the formula on the right-hand side may be simpliﬁed to maxi=0,... ,n |xi|. This is precisely the assertion.

1.10. Remark. Despite being so primitive, the naive height is actually suf- ﬁcient for most applications. For example, in the numerical experiments described in Part C, we will always work with the naive height.

Sec. 2] generalization to number fields 17

2. Generalization to number ﬁelds i. The deﬁnition.

2.1. Let K be a number ﬁeld. I.e., K is a ﬁnite extension of É.Itis well known from algebraic number theory [Cas67] that there is a set Val(K) of normalized valuations . ν on K satisfying the following conditions.

∈ É a) The functions . ν : K → Ê are indeed valuations. I.e., for x, y , i) x ν ≥ 0, ii) x ν =0if and only if x =0, iii) xy ν = x ν · y ν , iv) x + y ν ≤ x ν + y ν . b) There is the product formula x ν =1 ν∈Val(K) for every x ∈ K\{0}.

2.2. Further, for L/K adegreed extension of number ﬁelds, the sets Val(K) and Val(L) are compatible in the following sense. i) For every . μ ∈ Val(L), there are a valuation . ν ∈ Val(K) and dμ ∈ Æ such that | dμ . μ K = . ν .

In this case, it is said that . μ is lying above . ν . ii) For every . ν ∈ Val(K), there are only a ﬁnite number of valuations ∈ . μ1 ,..., . μl Val(L) lying above . ν . One has l

dμi = d. i=1

This implies l d x ν = x μi i=1 for every x ∈ K.

2.3. A valuation is called Archimedean if it lies above the valuation . ∞

of É. Otherwise, it is called non-Archimedean. If ν is non-Archimedean, then one has the ultrametric triangle inequality

x + y ν ≤ max{ x ν , y ν } .

18 the concept of a height [Chap. I

2.4. Definition. Let K be a number field of degree d. Then, for n (x0 : ... : xn) ∈ P (K), one puts 1 d Hnaive(x0 : ... : xn):= max xi ν . i=0,... ,n ν∈Val(K) This height function is the number field version of the naive height on Pn.Itis usually called the absolute height.

n 2.5. Lemma. Let K be a number ﬁeld, and let (x0 : ... : xn) ∈ P (K). Further, let L ⊃ K be a ﬁnite extension.

Then the absolute height Hnaive(x0 : ... : xn) remains unchanged when (x0 : ... : xn) is considered as an L-rational point. Proof. Put d := [L : K]. Then, by the properties of the valuations, we have

1 d Hnaive(x0 : ... : xn)= max xi ν i=0,... ,n ν∈Val(K) 1 dd = max xi μ i=0,... ,n ν∈Val(K) μ∈Val (L) μ above ν 1 dd = max xi μ , i=0,... ,n μ∈Val(L) which is exactly the formula for Hnaive(x0 : ... : xn) considered as an L-rational point.

2.6. Proposition (D. G. Northcott). Let B,D ∈ Ê.Then

n n

| ∈ É } {x ∈ P ( É) x P (K) for [K : ]

Proof. We may work with the number ﬁelds K of a ﬁxed degree d. For x, we choose homogeneous coordinates such that some coordinate equals 1. Then it is clear that, for every valuation . ν and every index i,wehave

max{ x0 ν ,..., xn ν }≥max{1, xi ν } .

Multiplying over all ν and taking the d-th root therefore shows

Hnaive(x0 : ... : xn) ≥ Hnaive(1 : xi) .

Hence, it suﬃces to verify that the set

1 1

| ∈ É } {(1 : x) ∈ P ( É) (1 : x) P (K) for [K : ]=d and Hnaive(1 : x)

Sec. 2] generalization to number fields 19

For this, we write σ1(x),...,σd(x) ∈ É for the elementary symmetric functions in the conjugates of x. According to Vieta, x is a zero of the polynomial

d d−1 d−2 d Fx(T ):=T − σ1(x)T + σ2(x)T + ... +(−1) σd(x) ∈ É[T ] .

Lemma 2.7 shows that d d H 1:σ (x) ≤ · H (1 : x)rd ≤ · Brd. naive r r naive r

Thus, by Fact 1.3, we know that for each σr(x) there are only finitely many possibilities. Hence, there are only finitely many possibilities for the polynomial Fx and, therefore, only finitely many possibilities for x. This completes the proof.

2.7. Lemma. Let K be number ﬁeld of degree d that is Galois over É. ∈ For x K,denotebyσ1(x), ... ,σd(x) ∈ É the elementary symmetric functions in the conjugates of x.Then d H 1:σ (x) ≤ · H (1 : x)rd. naive r r naive

Proof. We denote the conjugates of x by x1,...,xd.Letν be any valuation of K. Then σr(x) ν = xi · ... · xi 1 r 1≤i < ...

r max{1, σr(x) ν}≤C(d, r, ν) · max max{1, xi ν } i r ≤ C(d, r, ν) · max{1, xi ν } . i

Multiplying over all valuations of K yields d d d H 1:σ (x) ≤ · H (1 : x )rd naive r r naive i i d d 2 = · H (1 : x)rd r naive since conjugate points have the same height. This inequality is equivalent to the assertion.

20 the concept of a height [Chap. I

2.8. For most theoretical investigations and, perhaps, some practical ones, it is actually better to work with the logarithm of the absolute height.

n 2.9. Deﬁnition. For (x0 : ... : xn) ∈ P ( É), put

hnaive(x0 : ... : xn):=logHnaive(x0 : ... : xn) .

The height function hnaive is called the logarithmic height. ii. Application: The height deﬁned by an ample invertible sheaf.

2.10. Deﬁnition. Let X be a projective variety over a number ﬁeld K,and let L be an ample invertible sheaf on X.

Then a height function hL induced by L is given as follows. Let m ∈ Æ be such that L ⊗m is very ample. Put 1 hL (x):= h iL ⊗m (x) m naive N for x ∈ P a closed point. Here, iL ⊗m : P → P denotes a closed embedding deﬁned by L ⊗m.

2.11. Lemma. Let X be a projective variety over a number ﬁeld K,andlet (1) (2) L be an ample invertible sheaf on X.IfhL and hL are two height functions induced by L , then there is a constant C such that

(1) (2) | hL (x) − hL (x)|≤C

for every x ∈ X( É). Proof. First, it is clear that the k-tuple embedding fulﬁlls hnaive ρk(x) = k·hnaive(x)

N ⊗m

∈ É L for every k ∈ Æ and x P ( ). Hence, if hL is defined using , then the same height function may be defined using L ⊗km. (1) (2) We may therefore assume that hL and hL are defined using the same tensor L (1) (2) power of . Then the two embeddings iL ⊗m and iL ⊗m differ by an automorphism ι: PN → PN . We have to verify that | hnaive ι(x) − hnaive(x)|≤C

∈ N for every x P ( É). ι is explicitly given by

→ (x0 : ... : xN ) (x0 : ... : xN )

:= ((a00x0 + ...+ a0N xN ):... :(aN0x0 + ...+ aNNxN )) for some a =(aij)ij ∈ MN+1(K).

Sec. 3] geometric interpretation 21

For all but ﬁnitely many ν ∈ Val(K),wehavethatν is non-Archimedean and aij ν =1for all i and j. This yields

≤ max xi w max xi w i=0,... ,N i=0,... ,N for every number ﬁeld L and every w ∈ Val(L) lying above such a valuation.

For every other valuation of K, we ﬁnd a constant Dν such that

≤ dw/ν · max xi w Dν max xi w . i=0,... ,N i=0,... ,N The desired inequality hnaive ι(x) − hnaive(x) ≤ log Dν ν∈Val(K) follows immediately from this. For the inequality the other way round, one may work with ι−1 instead of ι.

3. Geometric interpretation

3.1. Heights are more closely related to modern algebraic geometry than this might seem from the deﬁnitions given in the sections above. The geometric interpretation of the concept of a height is the starting point of arithmetic intersection theory, a fascinating theory, which we will only touch upon here.

3.2. We return to the assumption that the ground ﬁeld is K = É.Thisis done mainly in order to ease notation. The theory would work equally well over an arbitrary number ﬁeld.

3.3. Deﬁnition. An arithmetic variety is an integral scheme that is projec-

tive and ﬂat over Spec .

3.4. Deﬁnition. A hermitian line bundle on an arithmetic variety X is a pair (L , . ) consisting of an invertible sheaf L ∈ Pic(X) and a continuous hermit-

L L X 

ian metric on the line bundle associated with on the complex space ( ). All hermitian line bundles on X form an abelian group, which is denoted by 0 Pic C (X ). For the group of all smooth (C∞) hermitian line bundles, one writes Pic (X ). Pic (X ) is usually called the arithmetic Picard group.

3.5. Example. The projective space P over Spec is an arithmetic vari-

ety. On Pn , there is the tautological invertible sheaf O(1). It is generated by the global sections X ,...,X ∈ Γ(Pn , O(1)).

0 n O O

Let us show two explicit hermitian metrics on (1) making (1) intoahermitian line bundle.

22 the concept of a height [Chap. I i) The Fubini–Study metric is given by 1 FS := 2 2 X0 Xn + ... + ∈ n O

for Γ(P , (1) ). ii) The minimum metric is given by min := min . i=0,... ,n Xi

In the case of the Fubini–Study metric, the formula should be interpreted in n such a way that FS(x)=0at all points x ∈ P ( ) where vanishes. Simi- larly, in the deﬁnition of min(x), the minimum is actually to be taken over all i such that xi =0 . The Fubini–Study metric is smooth (C∞). The minimum metric is continuous but not even C1.

3.6. Deﬁnition (Arakelov degree). For a hermitian line bundle (L , . )

on Spec , deﬁne its Arakelov degree by deg (L , . ):=log#(L /sL ) − log s

for s ∈ Γ(Spec , ) a non-zero section.

L L

3.7. Remark. is associated with a free -module L of rank one. is

 ⊗ ∈

the -vector space L . Thus, we work with a non-zero element s L and with the norm s ⊗ 1 . The deﬁnition is independent of the choice of s since

#(L /nsL )=n·#(L /sL ) , therefore log #(L /nsL ) = log #(L /sL ) + log n. On the other hand, log ns =log s +logn.

3.8. Fact. Let L be a hermitian line bundle on Spec .Ifthereisasec- ∈ L L tion s Γ(Spec , ) of norm less than one, then deg > 0. Proof. This follows immediately from the deﬁnition.

3.9. Fact. For two hermitian line bundles L1, L2 on Spec , one has deg (L1 ⊗ L2)=deg L1 + deg L2 .

Proof. Apply the deﬁnition to arbitrary non-zero sections s1 ∈ Γ(Spec , 1),

L ⊗ ∈ L ⊗L s2 ∈ Γ(Spec , 2), and to their tensor product s1 s2 Γ(Spec , 1 2).

Sec. 3] geometric interpretation 23

3.10. A É-rational point on the projective space is actually a morphism → n

x: Spec É P of schemes. The valuative criterion for properness implies that

there is a unique extension to a morphism from Spec , x / n

Spec É 7P n n n ∩ n n n n x

Spec .

3.11. Observation (Arakelov). Let x ∈ P ( É),andlet

→ n

x: Spec P

be its extension to Spec .Then ∗ hnaive(x)=deg x (O(1), . min) .

Proof. Write x in coordinates as (x0 : ... : xn) for x0, ... ,xn ∈ . Then the n ∗ O O pullback of X ∈ Γ(P , O(1)) is x ∈ Γ(Spec , x ( (1))).Since (1) is generated

i i by the global sections Xi,wehave

∗

x (O(1)) = x0,...,xn =gcd(x0,...,xn) · .

 We choose an index i0 such that xi0 =0. Using this section, we obtain ∗ ∗ ∗ deg (x (O(1)), . )=log#x (O(1))/x x (O(1)) − log x i0 i0 x

− i0 =log#gcd(x0,...,xn) /xi0 log min i=0,... ,n xi xi0 xi =log +log max i=0,... ,n gcd(x0,...,xn) xi0 max |xi| i=0,... ,n =log , gcd(x0,...,xn) which is exactly the logarithmic height of x.

3.12. This motivates the following general deﬁnition. Deﬁnition. Let X be an arithmetic variety, and let L be a hermitian line bundle

on X . ∈ X É Then the height with respect to L of the É-rational point x ( ) is given by ∗

hL (x)=deg x L . → Here, x: Spec X is the extension of x to Spec .

3.13. Examples. i) Let X = Pn ,andletL be the invertible sheaf O(1) equipped with the minimum metric. Then, as shown in Observation 3.11, the height with respect to L is the naive height.

24 the concept of a height [Chap. I

ii) Let X = Pn ,andletL be the invertible sheaf O(1) equipped with the Fubini– Study metric. Then the height with respect to L is the l2-height, which is given by 2 2 hl2 (x0 : ... : xn):=log x0 + ... + xn for (x0 : ... : xn)=(x0 : ... : xn)

projective coordinates such that all xi are integers and gcd(x0,...,xn)=1.

3.14. Lemma. Let f : X → Y be a morphism of arithmetic varieties, and let L be a hermitian line bundle on Y . ∈ X Then, for every x ( É),

hf ∗L (x)=hL (f(x)) . ◦ Proof. Let x be the extension of x to Spec .Thenf x is the extension of f(x)

to Spec .Thus, ∗ ∗L ◦ ∗L hf ∗L (x)=deg x (f ) = deg (f x) = hL (f(x)) .

3.15. Proposition. Let X be an arithmetic variety. a) Let . 1 and . 2 be hermitian metrics on one and the same invertible sheaf L . Then there is a constant C such that

| − | h(L , . 1)(x) h(L , . 2)(x)

for every x ∈ X ( É).

b) Let L1 and L2 be two hermitian line bundles. Then, for every x ∈ X ( É),

h (x)=h (x)+h (x) . (L1⊗L2) L1 L2 c) Let L be a hermitian line bundle such that the underlying invertible sheaf is am-

∈ ∈ X É ple. Then, for every B Ê, there are only ﬁnitely many points x ( ) such that

hL (x)

Proof. a) For x the extension of x to Spec ,wehave − ∗L ∗ − ∗L ∗ h(L , . 1)(x) h(L , . 2)(x)=deg x , x . 1 deg x , x . 2 . ∈ ∗L Working with a non-zero section s Γ(Spec , x ), we see that the latter diﬀer- ence is equal to ∗ ∗ ∗ ∗ log #(x L /sx L ) − log s 1(x) − log #(x L /sx L ) − log s 2(x) . (x) =log 2 . . 1(x)

Sec. 4] the adelic picard group 25

Since X ( ) is compact, there is a positive constant D such that 1 . (x) < 2

b) Clearly, for every x ∈ X ( É), one has ∗ ∗ ∗ h (x)=deg x (L ⊗L ) = deg (x L )⊗(x L ) (L1⊗L2) 1 2 1 2 ∗ ∗ = deg (x L )+deg (x L )=h (x)+h (x) . 1 2 L1 L2

∈ L ⊗n c) There is some n Æ such that is very ample. Part b) shows that it suffices to verify the assertion for L ⊗n. Thus, we may assume that L is very ample. Let i: X → Pn be the closed embedding induced by L .ThenL = i∗O(1). It follows from Tietze’s Theorem that there is a hermitian metric on O(1) such L ∗O that = i (1).ThenhL (x)=hO(1)(i(x)). It, therefore, suffices to show n fundamental finiteness for the height function hO(1) on P ( É).

Part a) together with Observation 3.11 shows that hO(1) diﬀers from hnaive by a bounded summand. Fact 1.3 yields the assertion.

3.16. Remark. It should be noted that there is a strong formal analogy of the concept of a height on an arithmetic variety to the concept of a degree in algebraic geometry over a ground field. The only obvious difference is that the role of the sections of an invertible sheaf is now played by small sections, say, of norm less than one. Nevertheless, it seems that the height of a point is actually some sort of arithmetic intersection number. This is an idea that has been formalized first by S. Yu. Arakelov [Ara] for two- dimensional arithmetic varieties and later by H. Gillet and C. Soulé [G/S90]for arithmetic varieties of arbitrary dimension. We will not give any details on arithmetic intersection theory here as this is not formally necessary for an understanding of the next chapters. To get an impression, the reader is advised to consult the articles [G/S90, G/S92] of H. Gillet and C. Soulé, the textbook [S/A/B/K], and the references therein. The article [B/G/S] is a good starting point, as well. It explains, in particular, how to construct a height not only for points but for algebraic cycles. The particular case of the arithmetic intersection theory on a curve over a number field had been developed earlier. The articles of S. Yu. Arakelov [Ara]andG.Falt- ings [Fa84] present the point of view taken before around 1990, which is a bit different from today’s.

4. The adelic Picard group i. The local case. Metrics induced by a model.

4.1. Let K be an algebraically closed valuation ﬁeld. The cases we have in

É mind are K = Ép for a prime number p and K = ∞ = .

26 the concept of a height [Chap. I

4.2. Deﬁnition. Let X be a K-scheme. Then, by a metric on an invertible sheaf L ∈ Pic(X), we mean a system of K-norms on the K-vector spaces L (x) for x ∈ X(K). This means, to every point x ∈ X(K) there is associated a function L → . : (x) Ê+ such that i) ∀x ∈ X(K) ∀y ∈ L (x): y =0⇔ y =0, ii) ∀x ∈ X(K) ∀y ∈ L (x) ∀t ∈ K : ty = |t| y .

L 4.3. Remark. If K = ,thenametricon is the same as a (possibly discontinuous) hermitian metric.

4.4. Definition. Assume K to be non-Archimedean, and let OK be the ring of integers in K. Further, let X be a K-scheme, and let L ∈ Pic(X). Then, by a model of (X, L ), we mean a triple (X , L,n) consisting of a natural ∼ number n, a flat projective scheme π : X → OK such that XK = X,andan ∼ ⊗n invertible sheaf L ∈ Pic(X ) fulfilling L |X = L .

4.5. Example. Assume K to be non-Archimedean, let OK be the ring of integers in K,andletX be a K-schemeequippedwithaninvertiblesheafL . Then a model (X , L,n) of (X, L ) induces a metric . on L as follows. ∗ x ∈ X(K) has a unique extension x: Spec OK → X .Thenx L is a projective ⊗n ⊗n OK -module of rank one. Each l ∈ L (x) induces l ∈ L (x) and, therefore, a rational section of x∗L.Put

1 n l (x):= inf |a||a ∈ K, l ∈ a · x∗L . (∗)

4.6. Deﬁnition. The metric . given by (∗) is called the metric on L induced by the model (X , L,n).

4.7. Remark. Note here that OK is, in general, a non-discrete valuation ring. In particular, OK will usually be non-Noetherian. Nevertheless, projectivity includes being of finite type [EGA, Chapitre II, Défini- tion (5.5.2)]. This means, for the description of X , only a finite number of elements

from OK are needed. É In the particular case K = Ép, the group ν(K) is isomorphic to ( , +).Thus,for

any ﬁnite set {a1, ... ,as}⊂O ,thereexistsadiscrete valuation ring O ⊆ O É Ép p containing a1,...,as. By consequence, X isthebasechangeofsomeschemethatisprojectiveovera discrete valuation ring.

Sec. 4] the adelic picard group 27

4.8. Deﬁnition. Let K be an algebraically closed valuation ﬁeld. Assume K to be non-Archimedean. Then a metric . on L ∈ Pic(X) is called continuous, respectively bounded, if . = f · . for . a metric induced by some model and f a function on X(K) that is continuous or bounded, respectively.

4.9. Remark. If K = , then we adopt the concepts of bounded, continuous, and smooth metrics in their the usual meaning from complex geometry. Note that smooth metrics are continuous and that continuous metrics are automat-

ically bounded in the case K = .

ii. The global case. Adelically metrized invertible sheaves. ∈ Æ 4.10. Definition. Let X be a projective variety over É and m . 1 X Then, by a model of X over Spec [ m ],wemeanascheme that is projective 1 X and flat over Spec [ m ] such that the generic fiber of is isomorphic to X.

L ∈ 4.11. Deﬁnition. Let X be a projective variety over É and Pic(X) be an invertible sheaf. a) Then an adelic metric on L is a system

{ }

. = . ν ν∈Val( É)

of continuous and bounded metrics on L ∈ Pic(X ) such that É Éν ν

∈ É É i) for each ν Val( É),themetric . ν is Gal( ν / ν )-invariant,

∈ 1 X ii) for some m Æ, there exist a model of X over Spec [ m ],aninvertible sheaf L ∈ Pic(X ), and a natural number n such that

∼ ⊗n L |X = L

X L| and, for all prime numbers p m,themetric . νp is induced by ( p, Xp ,n). b) An invertible sheaf equipped with an adelic metric is called an adelically metrized invertible sheaf. c) All adelically metrized invertible sheaves on X form an abelian group, which will be called the adelic Picard group of X and denoted by Pic (X).

4.12. Notation. Let X be a model of X over Spec . Then taking the induced metric yields two natural homomorphisms iX : Pic(X ) → Pic(X) ,

X → ⊗ É →

aX :ker(Pic( ) Pic(X)) Pic(X) .

Further, one has the forgetful homomorphism

v : Pic (X) → Pic(X) .

28 the concept of a height [Chap. I

4.13. Notation. The models of X, together with all birational morphisms between them, form an inverse system of schemes. This is a ﬁltered inverse system since, for two models X and X , the closure of the diagonal

⊂ × ⊂ X × X Δ X Spec É X Spec projects to both of them. Thus, the arithmetic Picard groups Pic (X ) for all models X of X form a ﬁltered direct system. The injections Pic (X ) → Pic (X) ﬁt together to yield an injection

X → ιX :− lim→ Pic( ) Pic(X) .

Similarly, the usual Picard groups Pic(X ) form a ﬁltered direct system, too. We get a homomorphism

X → ⊗ É −→ αX :− lim→ ker(Pic( ) Pic(X)) Pic(X) .

4.14. Deﬁnition (Metric on v−1(L ) ⊆ Pic (X)). Let X be a projective va- L L riety over É.OnX,let( , . ) and ( , . ) be two adelically metrized invertible sheaves with the same underlying sheaf. Then the distance between (L , . ) and (L , . ) is given by L L δ(( , . ), ( , . )) := δν( . ν , . ν )

ν∈Val( É) for . ν (x) δν( . ν , . ν ):= sup log . . ν (x) x∈X( Éν ) 4.15. Lemma. δ is a metric on the set v−1(L ) of all metrizations of L . Proof. We have to show that the sum is always finite. For this, we note first that the metrics . ν and . ν are bounded by definition. Therefore, each summand is finite. We may thus ignore a finite set S of primes and assume that . and . are given by triples (X , L,n) and (X , L,n), respectively, in the sense of Defini- tion 4.11.a.ii). ∼ ∼ X −→= −→= X

The isomorphism É X may be extended to an open neigh- É bourhood of the generic∼ ﬁber. Therefore, enlarging S ifnecessary,wehavean X = \ X L isomorphism −→ X of schemes over Spec S. Further, the triple ( , ,n) may be replaced by (X , L⊗n ,nn) without any change of the induced metric. Thus, without restriction, n = n. To summarize, we are reduced to the case that . and . are given by (X , L,n) and (X , L,n). We have an isomorphism ∼ ∼ = ⊗n =

L |X −→ L −→ L |X ,

É É which may be extended to an open neighbourhood of the generic fiber. There- fore, in the definition of δ((L , . ), (L , . )), all the summands vanish, except finitely many. Positivity, symmetry, and the triangle inequality are clear.

Sec. 4] the adelic picard group 29

4.16. Remark. It is convenient to consider two adelically metrized invertible sheaves with diﬀerent underlying sheaves as of distance inﬁnity. Then the distance δ is no longer a metric but only a separated écart in the sense of N. Bour- baki [Bou-T, §1].

4.17. Lemma. Let f : X → Y be a morphism of projective varieties over É. i) Then the homomorphism f ∗ : Pic (Y ) → Pic (X) is continuous with respect to the metric topology. ii) Even more, ∗ ∗ δ(f L1,f L2) ≤ δ(L1, L2) for arbitrary adelically metrized invertible sheaves L1, L2 ∈ Pic(Y ). Proof. ii) is obvious. i) follows immediately from ii). iii. Adelic heights.

4.18. Example. Let K be a number ﬁeld. Then there is an isomorphism ∼

l : Pic(Spec K) = Ê im λ, w∈Val(K) where λ is the mapping ∗

λ: K −→ Ê , w∈Val(K)

t → (− log |t|w)w∈Val(K) .

Proof. We have Pic(Spec K)=0. Thus, only the metrizations of the trivial invertible sheaf OSpec K have to be considered. We choose a section 0 = s ∈ Γ(Spec K, O )=K. Spec K

∈ ∼ É ⊗ É For every ν Val( ), one has K É ν = w Kw [Cas67, formula (10.2)], where w runs through the valuations of K, extending ν. Hence,

× ∼ É

Spec K Spec É Spec ν = Spec Kw, w and accordingly for the structure sheaves. Thus, the section s induces the homomorphism −→ ι: Pic(Spec K) Ê , w∈Val(K) given by (L , { . ν}w∈Val(K)) → (− log s w)w∈Val(K). Here, condition ii) of Defi- nition 4.11.a) ensures that all the valuations of s, except finitely many, are actually equal to 1. Therefore, the image of ι is indeed contained in the direct sum. Furthermore, ι is a surjection, as the existence of an appropriate model is required only outside a finite number of primes.

30 the concept of a height [Chap. I

Finally, there is the ambiguity caused by the choice of the section s. Two non-zero ∈ O ∈ ∗ sections s, s Γ(Spec K, Spec K ) diﬀer by a factor t K . Thus, the correspond- − | | ing images in w Ê diﬀer by the summand ( log t w)w∈Val(K). The assertion follows.

4.19. Deﬁnition (Arithmetic degree). For an adelically metrized invertible sheaf (L , . ) on Spec K, deﬁne its arithmetic degree by deg (L , . ):=s(l(L , . )) .

→ Ê Here, s: w Ê im λ is the summation map.

4.20. Remarks. i) The product formula implies that the summation map s factors via w Ê im λ. → ii) The arithmetic degree is a group homomorphism deg : Pic(Spec K) Ê.

L ∈ L L iii) For every Pic(Spec ), one has deg (iSpec ( )) = deg ( ). Indeed, this is directly seen from the various deﬁnitions.

4.21. Remarks. The adelic Picard groups as defined here tend to be very large groups. They are not designed to be particularly interesting invariants for a purpose such as distinguishing between non-isomorphic varieties. They just give a general framework for the determination of an individual height function. This framework is in fact more flexible than the concept of a height with respect to a hermitian line bundle, introduced in the definition in Sub- section 3.12. Further, the height function defined by an ample adelically metrized invertible sheaf L differs only by a bounded function from that defined in a naive way by the underlying invertible sheaf L , cf. Definition 2.10.

4.22. Deﬁnition. Let X be a regular, projective variety over É,andlet L ∈ Pic (X) be an adelically metrized invertible sheaf. Then the absolute height with respect to L of an K-valued point x ∈ X(K) for K a number ﬁeld is given by 1 hL (x):= deg L |x .

[K : É]

4.23. Example. Consider the situation that X = Pn and L = O(1), É

equipped with the adelic metric, induced by the model (Pn , O(1), 1).

Then hL coincides with the naive height hnaive.

Proof. This is immediate from Deﬁnitions 4.22 and 4.6, combined with Deﬁni- tions 2.4 and 2.9.

Sec. 4] the adelic picard group 31

4.24. Lemma. Let f : X → Y be a morphism of projective varieties over É, and let L be an adelically metrized invertible sheaf on Y . Then, for every number ﬁeld K and every x ∈ X (K),

hf ∗L (x)=hL (f(x)) .

Proof. According to Deﬁnition 4.22, we have 1 ∗ ∗L 1 ◦ ∗L

hf ∗L (x)= deg x (f ) = deg (f x) = hL (f(x)) . É [K : É] [K : ]

4.25. Proposition. Let X be a projective variety over É. a) Let . 1 and . 2 be adelic metrics on one and the same invertible sheaf L . Then there is a constant C such that

| − | h(L , . 1)(x) h(L , . 2)(x)

h = h + h . (L1⊗L2) L1 L2 c) Let L be an adelically metrized invertible sheaf such that the underlying invert- ∈ ible sheaf is ample. Then, for every B,D Ê, there are only ﬁnitely many points ∈ x X(K) such that [K : É]

hL (x)

1 Proof. a) Over some Spec [ ], the adelic metrics . 1 and . 2 are induced m by models (X1, L1,n1) and (X2, L2,n2), respectively. We may assume without restriction that n1 = n2 =: n,asamodel(X , L ,n) may always be replaced by (X , L⊗n ,nn) without change. _ _/ Further, there is a birational equivalence ι: X X that extends the identity _ _/ 1 2 map on X. It defines an isomorphism U1 U2 between suitable open subschemes completely containing the generic fibers. As X1 and X2 are proper, the complementary closed subsets are contained in finitely many special fibers. Hence, ι induces 1 an isomorphism over some Spec [ mm ] for m =0. As an analogous argument applies to the invertible sheaves L1 and L2,weseethat . 1 and . 2 coincide up to finitely many primes ν1, ... ,νk. Further, an adelic metric consists of bounded metrics. Hence, there is a constant D such that 1 . ≤ . ≤ D . D 1,νi 2,νi 1,νi for i =1, ... ,k.

32 the concept of a height [Chap. I

For x ∈ X(K),wenowhave 1 ∗ ∗ 1 ∗ ∗

h(L , . )(x) − h(L , . )(x)= deg x L ,x . 1 − deg x L ,x . 2 É 1 2 [K: É] [K: ] 1 ∗ −1 = deg OSpec K ,x ( . 1 ⊗ . ) . [K: É] 2

Working with the non-zero section 1 ∈ Γ(Spec K, OSpec K ), we see that the latter expression is equal to k 1 1

− log 1 w = − log 1 w É [K: É] [K: ] w∈Val(K) i=1 w|νi

−1 ⊗ É for . w the extension of ( . 1 . 2 )ν from ν to Kw. This includes a raise to the [Kw : Éν ]-th power [Cas67, Sec. 11]. Hence, k | 1 h(L , . )(x) − h(L , . )(x)|≤ [Kw : Éν ]logD = k log D, 1 2 [K: É] |

i=1 w νi É when we observe the fact that [Kw : Éν ]=[K : ]. This is the assertion. w|νi b) Clearly, for every x ∈ X (K), one has 1 ∗ 1 ∗ ∗

h L ⊗L (x)= deg x (L1 ⊗L2) = deg (x L1)⊗(x L2) É ( 1 2) [K: É] [K: ] 1 ∗ 1 ∗

= deg (x L1)+ deg (x L2)=hL (x)+hL (x) . É [K: É] [K: ] 1 2

L ⊗k c) There is some k ∈ Æ such that is very ample. Part b) shows that it suﬃces to verify the assertion for L ⊗k. Thus, we may assume that L is very ample. Let i: X → PN be the closed embedding induced by L .ThenL = i∗O(1). É O Tietze’s Theorem shows that there exists a hermitian metric . on (1)PN such ∗ that . L ,∞ = i . . X L 1 Further, there is the model ( , ,n) over some Spec [ m ], which induces the N

adelic metric . L outside the primes dividing m. The morphism i: X → P É _ _/ N extends to a rational map i : X P 1 , the locus of indeterminacy of which is a

[ ] closed subset of X not meeting the genericm fiber. As X is proper, this shows that 1 i is actually a morphism over Spec [ mm ], for a certain m =0. Now, by [EGA, Chapitre III, Proposition (4.4.1)] together with [EGA, Chapit- re III, Corollaire (4.4.9)], there is an open subset U ⊆ X containing the generic fiber such that i |U is an embedding. Again as X is proper, we see that i is an 1 embedding over Spec [ mm ] for some m =0. Define on O(1) an adelic metric in such a way that, at the finite primes ν dividing N mm , it extends the metric defined by . L ,ν on i(X) ⊆ P and, at the remaining N finite primes, it is induced by the model (P 1 , O(1), 1). Further, at the infinite

[ ] mm prime, we let it agree with the hermitian metric from above. ∗O L Then i (1) = . Consequently, hL (x)=hO(1)(i(x)). It is, therefore, suﬃcient N to show the assertion for the height function hO(1) on P .

But part a) together with Example 4.23 shows that hO(1) diﬀers from hnaive by a bounded summand. Thus, Proposition 2.6 implies the assertion.

Sec. 4] the adelic picard group 33

4.26. Remarks. a) The concept of an adelically metrized invertible sheaf as described here is essentially that introduced by V. V. Batyrev and Yu. I. Manin in [B/M]. It was used, for example, by E. Peyre in [Pe02]. A small diﬀerence is that Batyrev and Manin ﬁx norms on the spaces L (x) only

∈ ∈ É for x X( Éν ), not for x X( ν ). Correspondingly, their concept of adelic metric leads to a height for points rational over the base field and not to an absolute height. b) There is a somewhat different concept of an adelic Picard group, which is due to S. Zhang [Zh95b]. His definition leads to by far smaller groups. In fact, the adelically metrized invertible sheaves that are globally induced by a model are topologically dense in PicZh(X). This has the following consequence, which is highly interesting for many applica-

tions. There is a continuous -multilinear map

× × −→ Ê Pic Zh(X) ... PicZh(X) , dim X+1 times which is called the adelic intersection product. It is uniquely determined by the condition that it agrees with the arithmetic intersection product of H. Gillet and C. Soulé [G/S90, Theorem 4.2.3] when restricted to adelically metrized invertible sheaves induced by models. More details are given in [Zh95b].

CHAPTER II

Conjectures on the asymptotics of points of bounded height

I have no satisfaction in formulas unless I feel their numerical magni- tude. William Thomson 1st Baron Kelvin, (Life (1943) by Sylvanus Thompson, p. 827)

1. A heuristic

1.1. Let X be a projective variety over É. Then one of the most natural questions to ask is

(∗)DoesX have ﬁnitely many or inﬁnitely many É-rational points?

∞ 1.2. If the answer is that #X( É) < , then one might ask for the pre-

cise number. Otherwise, if X( É) is inﬁnite, then one may ask for the asymptotics

of the rational points with respect to a certain height function H on X( É).

(†) What is the asymptotics of the function NX,H given by

| } NX,H(B):=#{x ∈ X( É) H(x)

for B →∞?

1.3. If X is a hypersurface of degree d in Pn, then there is a statistical heuristic for the asymptotics of NX,Hnaive .

1.4. Statistical heuristic. Let X be a hypersurface of degree d in Pn . É Then ∼ · n+1−d NX,Hnaive (B) C B for some positive constant C. n ∼ n+1 Proof. On P , the total number of É-rational points of height

(x0 : ... : xn)

∈ − ∼ n+1 for xi ∈ such that xi [ B,B]. There are B such (n +1)-tuples and the probability that gcd(x0, ... ,xn)=1is positive.

36 conjectures on points of bounded height [Chap. II

Further, X is given by a homogeneous form F of degree d. Its range of values on [−B,B]n+1 is, a priori, [−DBd,DBd] for a certain positive constant D. We assume that the values of F are equally distributed in that range. The value zero is then hit ∼Bn+1−d times.

1.5. Remark. This heuristic is deﬁnitely an interesting guideline about which behaviour to expect. It is, however, too naive in order to work well in all cases.

1.6. Example (Too few points—p-adic unsolvability). Consider the cubic

fourfold X in P5 given by the equation É

x3 +7y3 +49z3 +2u3 +14v3 +98w3 =0.

∅ É ∅ Then X( É7)= , which implies X( )= . Note that the statistical heuristic would predict cubic growth for the number of

É-rational points x on X such that Hnaive(x)

1.7. Example (Too few points—the Brauer–Manin obstruction). Consider

the cubic surface X in P3 given by the equation É 3 w(x + w)(12x + w)= x + θ(i)y +(θ(i))2z . i=1 Here,

i − ∈ É θ := 38 + ζ19 (ζ19)

∗ 3 ∈ i ( 19) (i)

and θ are the three conjugates of θ in É.

 ∅ É ∅

As will be shown in Example IV.5.24, X( É ) = but X( )= .OnX,thereis an obstruction to the Hasse principle, which is an instance of the general Brauer– Manin obstruction, cf. Section IV.2. In this case, the statistical heuristic would predict linear growth for the number

of É-rational points x on X such that Hnaive(x)

1.8. Examples (Too many points—accumulating subvarieties). i) Consider

the smooth threefold X ⊂ P4 given by É

x4 + y4 = z4 + v4 + w4 .

Then the statistical heuristic predicts linear growth. I.e., there should be ∼ CB rational points of height

a,1 É

3 3 3 3 3 Va,1 : ax = y + z + v + w .

Sec. 1] aheuristic 37

Here, quadratic growth is predicted by the statistical heuristic.

However, Va,1 clearly contains the lines given by v = −w and

(x : y : z)=(x0 : y0 : z0)

for (x0 : y0 : z0) a É-rational point on the curve

3 3 3 Ea : ax = y + z .

− ∈ This is a twisted Fermat cubic. One has the rational point (0:1:( 1)) Ea( É).

Therefore, Ea is an elliptic curve over É. The lines described form a cone over an ∼ 2 elliptic curve. There are B É-rational points of height

Selmer as early as 1951 [Sel]. For example, E6( É) is of rank one, generated by (21 : 17 : 37).

1.9. Remark. These examples grew out of our study of diagonal cubic and quartic threefolds, which is described in Chapter VI.

1.10. Geometric interpretation. The exponent n+1−d appearing in the statistical heuristic may be positive, zero, or negative. According to this distinction, there are three cases. This distinction creating three cases coincides perfectly well with the Kodaira clas- siﬁcation of projective varieties into Fano varieties, varieties of intermediate type, and varieties of general type. Indeed, the anticanonical divisor (−K) on a degree d hypersurface in Pn is exactly O(n +1− d).

1.11. Remark. The Kodaira classiﬁcation is a purely geometric one. It does not make use of any arithmetic information on the projective variety considered.

Only the complex algebraic variety produced by base change to is exploited. It is a very remarkable observation that whether a projective variety X is Fano, of intermediate type, or of general type seemingly has a lot of inﬂuence on the set of

É-rational points on X.

1.12. More concretely, expectations are as follows. First case. X is a variety of general type. By deﬁnition, this means that the canonical invertible sheaf K is ample. In the case of a hypersurface, this corresponds to the case that n +1− d<0. Here, the statistical heuristic is rather illogical. It states that for a large height bound, we expect fewer points than for a small one. In the limit for B →∞,we have Bn+1−d → 0.

One would therefore expect only very few É-rational points on X.Thisisexactly the content of the conjecture of Lang.

38 conjectures on points of bounded height [Chap. II

Second case. X is a variety of intermediate type. In the case of a hypersurface, this corresponds to the case that n +1− d =0. Here, the statistical heuristic states that there should be a constant number of points, independently of the height bound B.

One would therefore expect that there are a few É-rational points on X.Amore precise statement is given by the conjecture of Batyrev and Manin. Third case. X is a Fano variety. By deﬁnition, this means that the anticanonical invertible sheaf K ∨ is ample. In the case of a hypersurface, this corresponds to the case that n +1− d>0.

Here, one would expect that there are a lot of É-rational points on X. A by-far- more-precise formulation is given by the conjecture of Manin.

2. The conjecture of Lang

2.1. The conjecture of Lang deals with the case of a variety of general type. There are actually several versions of it [Lan86].

2.2. Conjecture (Lang). Let X be a smooth, projective variety of general type over a number field K.Then i) (Weak Lang conjecture) The set of rational points X(K) is not Zariski dense in X. ii) (Strong Lang conjecture) There is a Zariski closed subset Z ⊂ X such that, for any finite field extension L ⊇ K, one has that X(L)\Z(L) is finite.

2.3. Conjecture (Geometric Lang conjecture). Let X be a smooth, projective variety of general type over a ﬁeld K of characteristic 0. Then there is a proper Zariski closed subset Z(X) ⊂ X, called the Langian exceptional set, which is the union of all positive dimensional subvarieties that are not of general type.

2.4. Remark. These conjectures are strongly interrelated. If the geometric Lang conjecture is true, then the weak Lang conjecture implies the strong Lang conjecture.

2.5. Lang’s conjectures are known to be true when X is a subvariety of an abelian variety. This was proven by G. Faltings in [Fa91]. Faltings’ 1991 result includes the case that X is a curve of general type. I.e., a curve of genus g ≥ 2. This particular case of Lang’s conjecture has been popular for decades as the Mordell conjecture. The Mordell conjecture was proven by G. Faltings in 1983 [Fa83].

Sec. 2] the conjecture of lang 39

2.6. Examples. If X is a curve, then there is no Langian exceptional set Z. Weak and strong Lang conjectures are therefore equivalent. This is no longer the case for surfaces of general type. i) For example, the quintic surface given by

x5 + y5 + z5 + w5 =0

in P3 contains the line “x = −y, z = −w”, on which there are inﬁnitely many É

É-rational points. ii) Another example of a surface of general type containing a projective line is provided by the Godeaux surface [Bv, Example X.3.4]. iii) Let V 3 be the cubic threefold in P4 given by a,b É

ax3 = by3 + z3 + v3 + w3 .

Then the moduli space La,b of the lines on Va,b is a surface of general type [Cl/G, Lemma 10.13]. If the cubic curve 3 3 3 Ea,b : ax + by + z =0 contains a rational point, then, on Va,b, there are the lines given by v = −w

and (x : y : z)=(x0 : y0 : z0) for (x0 : y0 : z0) ∈ Ea,b( É). We call these lines the obvious lines on Va,b. ∅ In other words, if Ea,b( É) = , then the surface La,b, which is of general type, con-

tains a copy of the elliptic curve Ea,b. There are inﬁnitely many É-rational points on Ea,b, for example for a =6and b =1.

2.7. An experiment. We searched systematically for É-rational lines on the cubic threefolds Va,b for a, b =1, ... ,100, a ≥ b. Our method is described in detail in Chapter VI, Section 4. It guarantees that every line that contains a point of height <5 000 is certainly found.

The results may be interpreted as providing numerical evidence for Lang’s conjec- É ture. Indeed, a É-rational point on La,b corresponds to a -rational line on Va,b. Points on the elliptic curves lying on La,b correspond to the obvious lines. In agreement with Lang’s conjecture, only very few non-obvious lines were found. On all the varieties Va,b for a, b =1, ... ,100, a ≥ b, together, there are only 42 non- obvious lines containing a point of height <5 000. Each such line actually contains a point of height ≤15. We describe these lines explicitly in Chapter VI, Section 4, Ta- ble 1.

2.8. Remark. It is expected that Lang’s conjecture is true not only over

number fields but over every field that is finitely generated over É.Thisversion of Lang’s conjecture has a number of surprising consequences. The reader may get an impression of these in the article [A/V] of D. Abramovich and J. F. Voloch.

40 conjectures on points of bounded height [Chap. II

3. The conjecture of Batyrev and Manin i. Generalities.

3.1. Conjecture (V. V. Batyrev and Yu. I. Manin). Let X be a smooth, projective variety over a number ﬁeld k,andletL be an ample invertible sheaf on X. We denote by K and L divisors on X that represent the canonical invertible sheaf and L , respectively. ∈ Suppose r Ê is such that

∈ ⊗ Ê

[K + rL] NS(X)

is the class of an eﬀective Ê-divisor. Then, for every ε>0, there exists a Zariski open subset X◦ ⊆ X such that

◦ ◦ { ∈ | } r+ε NX ,HL (B)=# x X (k) HL (x)

3.2. Here, HL is the exponential of the height hL defined by L as introduced in Definition I.2.10. It is determined only up to a factor that is bounded above and below by positive constants. The conjecture is consistent with these changes of the height function since Br+ε and (CB)r+ε differ by a constant factor.

3.3. Remark. This conjecture was ﬁrst formulated by V. V. Batyrev and Yu. I. Manin in [B/M]. An excellent presentation may be found in the survey article [Pe02]byE.Peyre.

3.4. Fact (Varieties of general type). The conjecture of Batyrev and Manin implies the weak Lang conjecture. Proof. Indeed, if X is a variety of general type, then the canonical invertible sheaf K itself is ample and we may work with L := K .

Then K+(−1)L =0is an eﬀective Ê-divisor. Hence, for every ε>0, the conjecture of Batyrev and Manin yields that

◦ −1+ε #{x ∈ X (k) | HL (x)

3.5. Let X be a smooth Fano variety. Then the anticanonical invertible sheaf K ∨ is ample, and we may consider an anticanonical height, which is deﬁned by L := K ∨.

Sec. 3] the conjecture of batyrev and manin 41

3.6. Fact (Fano varieties). For X a smooth Fano variety, the conjecture of Batyrev and Manin yields that

◦ ◦ { ∈ | ∨ } 1+ε NX ,HK ∨ (B)=# x X (k) HK (x)

3.7. Remarks. i) In the case of a Fano hypersurface, this assertion ﬁts perfectly well with the statistical heuristic. ii) However, for Fano varieties, the conjecture of Manin describes the growth

◦ of NX ,HK ∨ much more precisely. The most interesting case of the conjecture of Batyrev and Manin is, therefore, that of a variety of intermediate type. ii. Varieties of intermediate type.

3.8. Let X be a smooth, projective, minimal surface of Kodaira dimension 0. Fix an ample invertible sheaf L ∈ Pic(X). Then the conjecture of Batyrev and Manin states that, for every ε>0,thereexists a Zariski open subset X◦ ⊂ X such that

◦ ◦ { ∈ | } ε NX ,HL (B)=# x X (k) HL (x)

Indeed, 12K is linearly equivalent to zero in any of the four cases of the Kodaira clas- siﬁcation. Hence,

∈ ⊗ Ê

[K] NS(X)

is the class of an eﬀective Ê-divisor on X.

3.9. Example. If X is an abelian variety, then the conjecture of Batyrev and Manin is true. Indeed, if the rank of X(k) is equal to r,then

r/2 ◦ ∼ · NX ,HL (B) C log B.

3.10. Remark (K3 surfaces—known results). On the other hand, for K3 surfaces, the conjecture of Batyrev and Manin is open. Weak versions of the conjecture have been established only in some particular cases. i) For special types of K3 surfaces, most notably for Kummer surfaces associated to a product of two elliptic curves, a particular result has been obtained by D. McKinnon [McK00]. If d is the minimal degree of a rational curve on X,then ◦ ◦ 2/d NX ,HL (B) B for X the complement of the union of all rational curves on X of degree d. ii) An estimate of the same type was established by H. Billard [Bill, Théorème 4.1] for K3 surfaces given as a smooth hypersurface of multidegree (2, 2, 2) in P1 × P1 × P1.

42 conjectures on points of bounded height [Chap. II

3.11. Remark. Actually, for K3 surfaces, the Batyrev–Manin conjecture is implied by a very general conjecture due to P. Vojta [McK11].

3.12. Observation. There are examples of K3 surfaces over a number field k that contain infinitely many rational curves defined over k [Bill, Sec. 3]. Let X be such a K3 surface, and let C1,C2, ... be rational curves on X.De- note their degrees by d1,d2, ... . Assume that the curves are listed in such a way that the degrees are in ascending order. Then, on ◦ \ \ \ Xl := X C1 ... Cl−1 ,

◦ ≥ 2/dl there are still cl B k-rational points to be expected. Indeed, Xl contains a non-empty, open subset of Cl. In other words, there is no way to choose a uniform Zariski open subset X◦ ⊂ X such that ◦ ◦ { ∈ | } ε NX ,HL (B)=# x X (k) HL (x) 0. In order to fulﬁll the conjecture of Batyrev and Manin, one therefore has to choose X◦ in dependence of ε.

3.13. Example. Consider the diagonal quartic surface X given in P3 by É the equation x4 +2y4 = z4 +4w4 .

± On this K3 surface, there are the É-rational points (1 : 0 : 1:0)and

(±1 484 801 : ±1 203 120 : ±1 169 407 : ±1 157 520) . É Thesearetheonly É-rational points on X known and the only -rational points of (naive) height less than 108 [EJ2, EJ3].

A systematic search for É-rational points on the K3 surface X is described in Chapter V.

3.14. Remark. Generally speaking, not much is known about the arithmetic of K3 surfaces. Nevertheless, in 1981, F. Bogomolov formulated a very opti- mistic conjecture.

3.15. Conjecture (F. Bogomolov, cf. [Bo/T]). Let X be a K3 surface over anumberﬁeldk. Then every k-rational point on X lies on rational curve C ⊂ X (deﬁned over the algebraic closure k).

3.16. Remark. In general, for varieties of intermediate type, it is known that the validity of the Batyrev-Manin conjecture extends from a variety to all its unramiﬁed coverings [Mo-ta, Proposition 5].

Sec. 3] the conjecture of batyrev and manin 43 iii. A somewhat diﬀerent example.

2 3.17. Example. Let X be the blowup of P in nine points P1, ... ,P9 in general position.

Denote by E := E1 + ... +E9 the sum of the corresponding nine exceptional lines. According to the Nakai–Moizhezon criterion, a divisor aL − bE is ample if and only if a>3b>0.Further,wehaveK = −3L + E. Let a and b be such that aL − bE is ample. Then the conjecture of Batyrev and Manin therefore asserts that, for every ε>0, there exists a Zariski open subset X◦ ⊂ X such that

◦ 3/a+ε NX ,HO(aL−bE) (B) B .

On the other hand, assuming aL − bE to be very ample for simplicity, the em- 2 N bedding ι:( X )P → P is given by homogeneous forms of degree a. Hence, ≤ · a HO(aL−bE) ι(x) c Hnaive(x) for some constant c, which shows the lower bound

◦ 3/a NX ,HO(aL−bE) (B)=Ω(B ) for any Zariski open subset X◦ ⊂ X.

3.18. On X, there are inﬁnitely many exceptional curves D such that 2

D = −1 and DK = −1. For every d ∈ Æ, there are ﬁnitely many of the type dL − a1E1 − ... − a9E9. Relative to aL − bE, their degrees are

(dL − a1E1 − ... − a9E9)(aL − bE)=da − b(a1 + ... + a9) = da − b(3d − 1) = d(a − 3b)+b.

Since a − 3b>0, this expression tends to inﬁnity for d →∞. Only ﬁnitely many of 2 3/a the exceptional curves are of degree < 3 a, which is equivalent to B rational points of height

3.19. Remark. What is interesting in this example is that X is a non- minimal surface, the minimal model of which is Fano. In particular, there are many rational points on X. The only diﬀerence to the Fano case might seem to be that there is no distinguished height, which could be used in order to count them. The anticanonical sheaf is not ample. However, there are, at least potentially, inﬁnitely many accumulating subvarieties in analogy to the observations made for some K3 surfaces.

44 conjectures on points of bounded height [Chap. II

◦ 3.20. Problem. Is it possible to describe the growth of NX ,HO(aL−bE) more accurately? At least for a particular choice of a and b? Is there a variant of Manin’s conjecture for P2 blownupinninepoints?

4. The conjecture of Manin i. Some general facts on the cohomology of Fano varieties.

4.1. Fact. Let X be a smooth, projective variety over a ﬁeld k of characteristic 0. Assume that X is Fano. Then

i H (X, OX )=0 for every i ≥ 1. Proof. Denote the canonical invertible sheaf on X by K . Then the vanishing theorem of Kodaira [G/H, Chapter 1, Section 2] ensures that

i K ⊗ L H (X, OX )=0 for every ample invertible sheaf L ∈ Pic(X).PutL := K ∨.

4.2. Lemma. Let k ⊆ be an algebraically closed ﬁeld of characteristic 0, and let X be a smooth, projective variety over k. Assume that X is Fano. Then, for every prime number l, 1 i) Hét(X, l)=0. ii) The ﬁrst Chern class induces an isomorphism ∼

= 2

−→ ⊗ Pic(X) l Hét(X, l(1)) .

Proof. Let ﬁrst i ∈ Æ be arbitrary. By [SGA4, Exp. XVI, Corollaire 1.6], we ∈ i ∼ i Æ k k have, for every k , Hét(X, μl ) = Hét(X ,μl ). Further, the comparison theo-

i ∼ i k

k rem [SGA4, Exp. XI, Théorème 4.4] shows that Hét(X ,μl ) = H (X( ), /l ). On the other hand, by the universal coeﬃcient theorem for cohomology [Sp, Chap. 5, Sec. 5, Theorem 10],

i k ∼ i k i+1 k

 ⊗ ⊕ 

H (X( ), /l ) = H (X( ), ) /l Tor1(H (X( ), ), /l )

∼ i k i+1

⊗ ⊕ k = H (X( ), ) /l H (X( ), )l .

i+1 i+1

→ Here, the transition maps H (X( ), )lk+1 H (X( ), )lk are given by multiplication by l. A ﬁnite composition of them is the zero map. This implies

i ∼ i k

 ⊗ Hét(X, l(1)) = ←lim− H (X( ), ) /l

 ⊗ ‡

= H (X( ), ) l . ( )

 O i) Since H (X( ), X( ))=0, the long exact cohomology sequence associated

‡ to the exponential sequence yields that H (X( ), )=0. Formula ( ) implies the claim.

Sec. 4] the conjecture of manin 45

1 2

 O O ii) Here, we use both, H (X( ), X( ))=0and H (X( ), X( ))=0. The long exact cohomology sequence associated to the exponential sequence then shows that

2 c1 : Pic(X) −→ H (X( ), ) is an isomorphism. Tensoring yields isomorphisms

⊗ k 2 k −→ ⊗ Pic(X) /l H (X( ), ) /l , and going over to the inverse limit implies the assertion.

4.3. Remarks. i) The ﬁrst Chern class in étale cohomology is deﬁned using the Kummer sequence. Recall that there is the commutative diagram / 2πi /O exp /O∗ / 0 X X 0

2πi · 1 · exp( n ) exp( n ) = · n / /O∗ ( ) /O∗ / 0 μn X X 0 showing that this agrees with the deﬁnition based on the exponential sequence. ii) The tensor product does not, in general, commute with inverse limits. How-

ever, for a ﬁnitely generated -module A,

k ∼ ⊗ ⊗ ←lim− A /l = A l is the l-adic completion of A [Mat, Theorem 8.7]. ii. Fixing a particular anticanonical height.

4.4. Remark. The anticanonical height HK ∨ on X is determined only up to a certain factor that is bounded above and below by positive constants. To be able to make any statement on the value of the constant τ, we have to ﬁx a particular height function.

4.5. For this, according to Deﬁnition I.4.22, it is necessary to choose an { } K ∨ adelic metric . = . ν ν∈Val( É) on . For the remainder of this section, we ﬁx such an adelic metric once and for all.

4.6. Deﬁnition. Put

HK ∨ (x):=exph(K ∨, . )(x) for h(K ∨, . ) the absolute height with respect to the adelically metrized invertible sheaf (K ∨, . ) ∈ Pic (X) in the sense of the deﬁnition in Subsection I.4.22. We will call this height function the anticanonical height deﬁned by the adelic metric given in 4.5.

46 conjectures on points of bounded height [Chap. II

4.7. Most height functions occurring in practice are a lot simpler than the general theory. For this reason, it is probably wise to recall an elementary particular case. Choose → N ∨ ∼ ∗

i) a projective embedding ι: X P such that K = ι O(d) for some d ∈ Æ, É and ∨

ii) a continuous hermitian metric . ∞ on K .

Then the topological closure of ι(X) in PN is an arithmetic variety X ,thatisa

model of X over Spec . Put

∨ HK (x):=exph(O(d)|X , . ∞)(x)

for h(O(d)|X , . ∞) the height function with respect to the hermitian line bundle C0 (O(d)|X , . ∞) ∈ Pic (X) in the sense of the Deﬁnition in Subsection I.3.12.

4.8. This height function corresponds to the adelic metric on K ∨,whichis given by the following construction. (Cf. Example I.4.5.) i) . ∞ is part of the data given. ii) For a prime number p,themetric . p is given as follows. ∈ O → X ∗O Let x Ép. There is a unique extension x: Spec of x.Thenx (d) is Ép a projective OK -module of rank one. Each l ∈ O(d)(x) induces a rational section of x∗O(d).Put l (x):=inf{| a||a ∈ K, l ∈ a · x∗O(d)} .

4.9. Remarks. i) To deﬁne the metric . p for a particular prime p,a L 1 model of X over Spec [ m ] for p m is suﬃcient. { } K ∨ X ii) For every adelic metric . = . ν ν∈Val( É) on , there exist a model

1 ∨

∈ Æ K X of X over Spec [ m ] for a certain m and an extension of to such that . νp is induced by that model for every p m. iii. The conjecture.

4.10. The conjecture of Manin deals with the anticanonical height HK ∨ on a Fano variety.

4.11. Conjecture (Manin). Let X be a smooth, projective variety over É. Assume that X is Fano. Then there exist a positive integer r,arealnumberτ, and a Zariski open subset X◦ ⊆ X such that

◦ r

◦ ∨

{ ∈ É | }∼ NX ,HK ∨ (B)=# x X ( ) HK (x)

Sec. 5] peyre’s constant i—the factor α 47

4.12. Remarks. i) The factor logrB is new in comparison with the statistical heuristic given in 1.4. It has been known for a long time that such a factor is, in general, necessary. In fact, J. Franke, Yu. I. Manin, and Y. Tschinkel [F/M/T] showed in 1989 that Manin’s conjecture becomes compatible with direct products of Fano varieties only when a suitable logrB-factor is added. ii) At least in the case that X is a surface, it is expected that r = rk Pic(X) − 1. There are, however, counterexamples to this formula in dimension three [Ba/T96]. iii) On the other hand, in Chapter VI, we will present numerical evidence for Manin’s conjecture for diagonal cubic and quartic threefolds. For those, rk Pic(X)=1, and our experiments indicate that r =0.

4.13. Remarks. i) In this book, we shall not prove Manin’s conjecture in any non-trivial case. Nevertheless, Section 8 will contain some information on the methods of proof, working in particular situations, as they are available today. ii) In addition to the references given there, we recommend that the reader study the survey lecture of E. Bombieri [Bom], which originates from the Journées Arith- métiques at Edinburgh in 2007. It is concerned with various aspects of rational and algebraic points on algebraic varieties, including evidence for Manin’s conjecture, proven cases, and related questions.

5. Peyre’s constant I—the factor α

5.1. In [Pe95a], E. Peyre reﬁned Manin’s conjecture by providing an explicit description for the value of the coeﬃcient τ. Peyre’s constant is a product of several factors, which we will subsequently explain.

5.2. Deﬁnition (Cf. [Pe95a, Déﬁnition 2.4]). Let X be a projective alge-

braic variety over É. Choose an isomorphism ∼ = t ι: Pic(X)/ Pic(X)tors −→ .

⊗ t Ê Ê

Identify Pic(X) with according to ι.

⊂ ⊗ Ê Ê

Further, let Λeﬀ (X) Pic(X) = be the cone generated by the eﬀective di-

∨ t ∨ ⊂ Ê visors. Consider the dual cone Λeﬀ (X) ( ) .Then · { ∈ ∨ | − ≤ } α(X):=t vol x Λeﬀ x, K 1 .

 t ∨ t × Ê → Ê Here, ., . denotes the tautological pairing ., . :(Ê ) , and vol is the t ∨ ordinary Lebesgue measure on ( Ê ) .

5.3. Remark. In [Pe/T, Definition 2.5], α(X) is defined by an integral. An elementary calculation shows that the two definitions are equivalent.

∈ 5.4. Example. Suppose Pic(X)= . Denote by [L] Pic(X) the am- ∈ − ple generator. Let δ Æ be such that [ K]=δ[L].Thenα(X)=1/δ. In particular, one has α(X)=1for every smooth cubic surface such that rk Pic(X)=1. Indeed, (−K)2 =3is square-free. Therefore, [−K] ∈ Pic(X) is not divisible.

48 conjectures on points of bounded height [Chap. II

1 1 1 1 ⊕ 5.5. Example. α(P × P )=1/4. Indeed, Pic(P × P )=L1 L2. The eﬀective cone is generated by L1 and L2. In the dual space,

∨ 1 × 1 { | ≥ } Λeﬀ (P P )= al1 + bl2 a, b 0 .

Further, −K =2L1 +2L2. The condition x, −K≤1 is therefore equivalent to 2a +2b ≤ 1. The area of the triangle with vertices (0, 0), (1/2, 0),and(0, 1/2) is equal to 1/8.

5.6. Example. Let X be the blowup of P in a É-rational point.

Then α(X)=1/6.

⊕ − Here, Pic(X)= L E. The eﬀective cone is generated by E and L E.Inthe dual space, ∨ { | ≥ − ≥ } Λeﬀ (X)= al + be b 0,a b 0 .

Further, −K =3L − E. The condition x, −K≤1 is equivalent to 3a − b ≤ 1. The area of the triangle with vertices (0, 0), (1/3, 0),and(1/2, 1/2) is equal to 1/12.

5.7. Example. Let X be the blowup of P in six É-rational points that É

form an orbit under Gal( É/ ).Thenα(X)=4/3.

⊕ − Again, Pic(X)= L E. Here, the eﬀective cone is generated by E, L 1/3E, and 2L − 5/6E. I.e., by E and L − 5/12E. In the dual space,

∨ { | ≥ − ≥ } Λeﬀ (X)= al + be b 0,a 5/12b 0 .

Further, −K =3L − E. The condition x, −K≤1 is therefore equivalent to 3a − b ≤ 1. The area of the triangle with vertices (0, 0), (1/3, 0),and(5/3, 4) is equal to 2/3.

5.8. Example. Let X be a smooth cubic surface over É. Assume the É orbit lengths of the 27 lines under the Gal( É/ )-operation are [1, 10, 16]. Then α(X)=1.

Note that this is the generic case of a cubic surface containing a É-rational line. A computation using GAP, which we will discuss in paragraph III.8.23, shows

that rk Pic(X)=2. Compare the list given in the appendix. We claim that

⊕ É Pic(X)= K E for E the -rational line. Indeed, K and E are linearly independent since K2 =3and E2 = −1.Further, ∈ − ∈ if aK + bE Pic(X), then intersecting with K shows that 3a b while

− ∈ ∈ intersecting with a line skew to E shows a . Altogether, a, b .

Write D1 for the sum of the ten lines meeting E and D2 for the sum of the sixteen lines skew to E.AsD1K = −10 and D1E =10,wehaveD1 = −5K − 5E. Similarly, D2K = −16 and D2E =0imply D2 = −4K +4E.

The eﬀective cone is generated by E, D1,andD2. The calculations show that E and −K − E form a simpler system of generators.

Sec. 5] peyre’s constant i—the factor α 49

In the dual space,

∨ { | ≥ − − ≥ } Λeﬀ (X)= ak + be b 0, a b 0 .

Further, the condition x, −K≤1 is equivalent to −a ≤ 1. The area of the triangle with vertices (0, 0), (−1, 0),and(−1, 1) is equal to 1/2.

5.9. Example. Let X be a smooth cubic surface over É. Assume the É orbit lengths of the 27 lines under the Gal( É/ )-operation are [2, 5, 10, 10]. Then α(X)=3/2. Note that the two lines conjugate to each other are skew. Indeed, if they were con-

tained in a plane, then the third line contained in that plane would be É-rational. This example describes the generic case of a cubic surface containing two skew lines that are conjugate to each other over a quadratic number ﬁeld.

Again, the computation using GAP discussed in III.8.23 shows that rk Pic(X)=2. ⊕ We claim that Pic(X)= K E for E := E1 + E2, the sum of the two lines conjugate to each other. Indeed, the discriminant of the lattice spanned by K and E is −2 −2 = −10 . −23 As this number is non-zero, we see that K and E are linearly independent. Since (−10) is a square-free integer, the lattice may not be reﬁned.

Write D1 for the sum of the ﬁve lines meeting E1 and E2, D2 for the sum of the ten other lines meeting E1 or E2,andD3 for the sum of the ten lines not meeting E at all. Then, D1K = −5 and D1E =10imply that D1 = −3K − 2E. Simi- larly, D2K = −10 and D2E =10show that D2 = −4K − E. Finally, D3K = −10, D3E =0,andD3 = −2K +2E.

The eﬀective cone is generated by E, D1, D2,andD3. The calculations yield E and −3K − 2E as a simpler system of generators. In the dual space,

∨ { | ≥ − − ≥ } Λeﬀ (X)= ak + be b 0, 3a 2b 0 .

Further, the condition x, −K≤1 is equivalent to −a ≤ 1. The area of the triangle with vertices (0, 0), (−1, 0),and(−1, 3/2) is 3/4.

5.10. Remarks. i) Let X be a smooth cubic surface over É containing a

É-rational line. Assume that rk Pic(X)=2.Thenα(X)=1. ⊕

Indeed, the arguments given in Example 5.8 still show that Pic(X)= K E. É On the other hand, the Gal( É/ )-orbits of the 27 lines might break into smaller pieces. For example, assume there is an orbit consisting of k of the ten lines meeting E. This causes another generator D1 of the eﬀective cone. How- − ever, we have that D1K = k and D1E = k. Therefore, k k k D = − K − E = D 1 2 2 10 1 is simply a scalar multiple of D1.

50 conjectures on points of bounded height [Chap. II

Analogously, an orbit consisting of some of the sixteen lines skew to E leads to a ∨ scalar multiple of D2. This ensures that Λeﬀ (X) is the same as in the generic case described in Example 5.8.

ii) Similarly, let X be any smooth cubic surface over É containing two skew lines conjugate over a quadratic number ﬁeld. Assume that rk Pic(X)=2. Then α(X)=3/2.

5.11. Example. Let X be a smooth cubic surface over the rationals such

that rk Pic(X)=7. I.e., such that each of the 27 lines on X is deﬁned over É. 1 Then α(X)= 120 . This has been proven in 2006 in the Ph.D. thesis of U. Derenthal [Der,Theo- rem 8.3].

5.12. Remark. For smooth cubic surfaces in general, the value of α de- É pends only on the orbit structure of the 27 lines under the Gal( É/ )-operation. In particular, there are only 350 cases corresponding to the conjugacy classes of subgroups of W (E6). The examples given above actually cover the lion’s share of these cases. i) In fact, 137 of the 350 conjugacy classes of subgroups of W (E6) lead to Pi- card rank one. Then α(X)=1, as shown in Example 5.4. ii) Further, 133 conjugacy classes yield Picard rank two. The argument of Re- mark 5.10.i) alone covers 98 of them. Eight further conjugacy classes are treated by Remark 5.10.ii).

5.13. Remark. The calculations indicate that it might be possible to eﬀec- tively compute α(X) for each of the 350 conjugacy classes of subgroups of W (E6) that could appear as the Galois groups acting on the 27 lines. This has indeed been done very recently. In the cases of high Picard rank, the computation of the volumes of the resulting polytopes is a not a trivial matter. For example, László Lovász [Lov] explains that a good algorithm for the exact computation of the volume of a high-dimensional polytope is impossible. His suggestion is to use a Monte Carlo method instead. It turned out, however, that the dimensions were not yet critical, even for Del Pezzo surfaces of degree one, and that the software polymake could handle all the cases [D/E/J].

6. Peyre’s constant II—other factors i. The factor β.

6.1. Deﬁnition. Let X be a projective variety over É.Thenβ(X) is de- ﬁned as

1 É

β(X):=#H Gal( É/ ), Pic(X ) . É

Sec. 6] peyre’s constant ii—other factors 51

1 É

6.2. Remarks. We will study H Gal( É/ ), Pic(X ) in Chapter III. É Note the following facts.

i) If Pic(X )=,thenβ(X)=1. É ii) In the particular case that X is a smooth cubic surface, β(X) depends only on the

⊆ É conjugacy class of the subgroup G W (E6) deﬁned by the operation of Gal( É/ ) on the 27 lines. Actually, β(X) depends only on the decomposition of the 27 lines into Galois orbits.

We computed β(X) for each of the 350 conjugacy classes of subgroups of W (E6). The possible values are β(X)=1, 2, 3, 4,and9. More details are given in III.8.23, and the list is presented in the appendix. ii. The value of L at 1.

6.3. Lemma. Let X be a projective algebraic variety over É.Denoteby

É É ⊗ 

L( ., χPic(X )) the Artin L-function of the Gal( / )-representation Pic(X ) .

É É Then, for t the Picard rank of X,

t lim (s − 1) L(s, χPic(X )) s→1 É

is a real number diﬀerent from zero.

É É ⊗ 

Proof. The Gal( / )-representation Pic(X ) contains the trivial represen- É t

tation t times as a direct summand. Therefore, L(s, χPic(X ))=ζ(s) · L(s, χP ) É and t lim (s − 1) L(s, χPic(X ))=L(1,χP ) .

s→1 É É Here, ζ denotes the Riemann zeta function and P is a Gal( É/ )-representation that does not contain trivial components. [Mu-y, Corollary 11.5 and Corollary 11.4] show that L( ., χP ) has neither a pole nor a zero at 1. iii. The p-adic measures.

6.4. Deﬁnition. Let p be a prime number, and let x ∈ X( Ép) be arbitrary. × É

Choose local coordinates t1,...,tn on X Spec É Spec p in a neighbourhood of x.

These deﬁne a morphism of Ép-schemes

ι : U −→ An x x Ép from a Zariski open neighbourhood Ux of x such that ι(x)=(0, ... ,0) and ι is étale at x. −1 ∂ ∂ The tensor ﬁeld (ι )∗ ∧ ... ∧ is the restriction to the p-adic points of x,Ép ∂t1 ∂tn a section of the anticanonical sheaf K ∨ . Ép In a neighbourhood of x, we deﬁne the measure ωp by −1 ∂ ∂ ∗

(ι )∗ ∧ ... ∧ · ι (dt · ... · dt ) . É x,Ép x, p 1 n ∂t1 ∂tn

Here, each dti denotes a copy of the Haar measure [DieuII, Chap. XIV, Sec. 1]

on Ép normalized in the usual manner [Cas67, formula (7.2)].

52 conjectures on points of bounded height [Chap. II

6.5. Remark. This deﬁnition is independent of the choice of the coordinates t1,...,tn. Indeed, for t1,...,tn forming another system of local coordinates near x,wehave · · ∂(t1,...,tn) · dt1 ... dtn =det dt1 ...dtn ∂(t1,...,tn) and ∂ ∧ ∧ ∂ ∂(t1,...,tn) · ∂ ∧ ∧ ∂ ... =det ... . ∂t1 ∂tn ∂(t1,...,tn) ∂t1 ∂tn

6.6. Lemma. For a prime number p, suppose that the metric . p is induced X by a model of X. Then the measure ωp on X( Ép) is given as follows.

k Let a ∈ X ( /p ),andput

(k) k

{ ∈ É | ≡ } U a := x X( p) x a (mod p ) .

Then

n k

{ ∈ X | X ≡ } (k) # y ( /p ) y lifts to ( p),y a (mod p ) ωp(Ua ) = lim . n→∞ pn dim X Proof. This is simply a more concrete reformulation of the abstract deﬁnition.

6.7. Corollary. Let p be a prime such that X is smooth over p.Then,for

∈ X every a ( /p ), 1 ω (U(1))= . p a pdim X Proof. This is a consequence of Hensel’s lemma.

6.8. Remark. ωp may as well be described in terms of the aﬃne cone over X. For example, if X is a hypersurface of degree d in Pn, then one has − −k

1 p n (dim X+1)n

· ωp X( Ép) = lim #CX ( /p )/p 1 − p−1 n→∞ for k := n +1− d. A proof is given in [Pe/T, Corollary 3.5].

6.9. Deﬁnition (The measure τp on X( Ép)). Let p be a prime number.

We deﬁne the measure τp on X( Ép) by −1 Ip

τp := det 1 − p Frobp | Pic(X ) · ωp . É

Here, Pic(X )Ip denotes the ﬁxed module under the inertia group. É

6.10. Remark. In the easiest case that Pic(X )=, one has É 1 −1 Ip det 1 − p Frobp | Pic(X ) =1− . É p

Sec. 6] peyre’s constant ii—other factors 53

iv. The real measure. ∈ Ê

6.11. Deﬁnition (The measure τ∞ on X( Ê)). Let x X( ) be arbitrary. × Ê

Choose local coordinates t1, ... ,tn on X Spec É Spec in a neighbourhood of x.

These deﬁne a morphism of Ê-schemes

ι : U −→ An

x x Ê from a Zariski open neighbourhood Ux of x such that ι(x)=(0, ... ,0) and ι is

Ê → Ê étale at x.Consequently, ιx,Ê : Ux( ) is a diﬀeomorphism near x. −1 ∂ ∂ The tensor ﬁeld (ι )∗ ∧ ... ∧ is the restriction to the real points of a x,Ê ∂t1 ∂tn

section of the anticanonical sheaf K ∨. In a neighbourhood of x, we deﬁne the measure τ∞ by the diﬀerential form −1 ∂ ∂ ∗

(ι )∗ ∧ ... ∧ · ι (dt ∧ ... ∧ dt ) . Ê x,Ê x, 1 n ∂t1 ∂tn 6.12. Remark. Again, this deﬁnition is independent of the choice of the coordinates t1, ... ,tn.Ift1, ... ,tn form another system of local coordinates near x,then ∧ ∧ ∂(t1,...,tn) · ∧ ∧ dt1 ... dtn =det dt1 ... dtn ∂(t1,...,tn) and ∂ ∧ ∧ ∂ ∂(t1,...,tn) · ∂ ∧ ∧ ∂ ... =det ... . ∂t1 ∂tn ∂(t1,...,tn) ∂t1 ∂tn

6.13. In the situation that X is a hypersurface, the measure τ∞ may be described by far more concretely in terms of the Leray measure.

6.14. Deﬁnition. Let f ∈ Ê[X0, ... ,Xn] be a homogeneous polynomial such that (grad f)(x0, ... ,xn) =0 for every (x0, ... ,xn) diﬀerent from the origin. Then the Leray measure on “f =0” is given by the formula 1 ω = ω . Leray grad f hyp

Here, ωhyp denotes the usual hypersurface measure, cf. [DieuIII, Chap. XVI, Sec. 24, Problem 11].

6.15. Lemma. Let f ∈ Ê[X0, ... ,Xn] be as in Deﬁnition 6.14,andlet ⊂ n+1 ∂f U Ê be an open subset such that does not vanish on U. ∂x0 Then, on U ∩“f =0”, the Leray measure is given up to sign by the diﬀerential form 1 dx1 ∧ ... ∧ dxn . | ∂f/∂ x 0|

Proof. This is an immediate consequence of Sublemma 6.16 below.

54 conjectures on points of bounded height [Chap. II

n+1 → Ê 6.16. Sublemma. Let U ⊂ Ê be an open subset, and let f : U be a smooth function such that ∂f does not vanish. ∂x0 Then the hypersurface measure on “f =0” is given up to sign by the diﬀeren- tial form grad f dx1 ∧ ... ∧ dxn . | ∂f/∂ x 0|

Proof. It is well known that, having resolved the equation

f(x0, ... ,xn)=0 by x0, the hypersurface measure is given by the form 2 2 1+(∂x0/∂x1) + ... +(∂x0/∂xn) dx1 ∧ ... ∧ dxn . For every i =1, ... ,n, the equation f x0(x1, ... ,xn),x1, ... ,xn =0immedi- ∂f ∂x0 ∂f ately yields ∂x ∂x + ∂x =0. In other words, 0 i i ∂x ∂f ∂f 0 = − . ∂xi ∂xi ∂x0 Altogether, we ﬁnd the diﬀerential form n ∂f ∂f 2 1+ dx ∧ ... ∧ dx ∂x ∂x 1 n i=1 i 0 for the hypersurface measure. This is exactly the assertion.

6.17. Notation. Let X ⊆ Pn be a projective variety. We denote by CX É the aﬃne cone over X. O |

Ametric . on (1) X( ) deﬁnes a compact subset

|| |≤ | |≤ } N . := {x =(x0,...,xn) ∈ CX( Ê) x0 X0 (x),..., xn Xn (x) of the aﬃne cone that is symmetric to the origin. Note that the conditions | xi|≤ Xi (x) and | xj |≤ Xj (x) are equivalent to each other as long as xi,xj =0 .

O |

6.18. Examples. i) Consider on (1) X( ) the minimum metric . min

from Example I.3.5. Then

{ ∈ Ê || |≤ | |≤ } N . min = x =(x0, ... ,xn) CX( ) x0 1, ... , xn 1 is a hypercube.

Sec. 6] peyre’s constant ii—other factors 55 ii) Similarly, for the l2-metric,

| 2 2 ≤ } N = {x =(x , ... ,x ) ∈ CX( Ê) x + ... + x 1 . l2 0 n 0 n is the unit ball.

6.19. Proposition. Let F ∈ É[X0, ... ,Xn] be a homogeneous polynomial

of degree d and X ⊂ Pn be the hypersurface deﬁned by the equation F =0. É According to the adjunction formula, there is a canonical isomorphism ∼ O − −→= K ∨ ι: (n d +1) X .

∨ − 1 K 1 n+d−1 O | The metric . on X induces a metric ι . on (1) X( ).Let

N := N 1 ⊂ CX( Ê) ι−1 . n−d+1 − 1 be the subset deﬁned by ι 1 . n+d−1 . Then, in terms of N and the Leray measure, one may write n − d +1 τ∞(U)= ω 2 Leray CU∩N

for every measurable set U ⊆ X( Ê). Proof. First step. Explicit description of ι. The assertion is local in X. We may therefore assume without restriction that ∂F X0 =0 and =0 .Thent2, ... ,tn for ti := Xi/X0 form a local system of ∂X1 coordinates on X. d When putting F = X0 f(t1, ... ,tn), we ﬁnd

∂F d−1 ∂f = X0 . ∂X1 ∂t1 Under the Poincaré residue map [G/H, p. 147], the diﬀerential form df 1 ∂f ∧ dt2 ∧ ... ∧ dtn = dt1 ∧ ... ∧ dtn f f ∂t1 on Pn is mapped to dt ∧ ... ∧ dt . Dually, under this correspondence, the

2 n ∂ ∂ tensor field ∧ ... ∧ on X( ) is identified with ∂t2 ∂tn f ∂ ∧ ∧ ∂ 1 F ∂ ∧ ∧ ∂ ∂f ... = ∂F ... . ∂t1 ∂tn X0 ∂t1 ∂tn ∂t1 ∂X1 Furthermore, the Euler sequence identifies ∂ ∧ ... ∧ ∂ with the global section ∂t1 ∂tn n+1 ∈ n O X0 Γ P , (n +1) .

Altogether, −1 ∂ ∧ ∧ ∂ 1 n ι ... = ∂F X0 . ∂t2 ∂tn ∂X1

56 conjectures on points of bounded height [Chap. II

Second step. The measure τ∞ of Peyre.

Peyre’s measure τ∞ on X( Ê) is therefore given by the diﬀerential form 1 n d−1 ∂F n−d+1 X dt2 ∧ ... ∧ dtn = | X / |· X dt2 ∧ ... ∧ dtn . (§) ∂F 0 0 ∂X1 0 ∂X1

On the other hand, the subset N ⊂ CX( Ê) is given by

| x0|≤A := X0 .

Therefore, according to Fubini’s theorem, 1 ω = dx ∧ dx ∧ ... ∧ dx Leray | ∂F | 0 2 n ∂X CU∩N CU 1 |x0|≤A A xn−1 1 = 0 dx · dx ∧ ... ∧ dx d−1 0 | ∂F | 2 n x0 ∂X −A CU 1 x0=1 2 1 = · An−d+1 · dx ∧ ... ∧ dx . n − d +1 | ∂F | 2 n ∂X CU 1 x0=1 Note here that CX is an n-dimensional cone while the integrand is homogeneous of degree −(d − 1). Consequently, n − d +1 X n−d+1 ω = 0 dx ∧ ... ∧ dx , 2 Leray | ∂F | 2 n ∂X CU∩N CU 1 x0=1 which, according to formula (§), is exactly equal to τ∞(U).

6.20. Remarks. i) This result allows a generalization to complete intersections. ii) If X is a cubic surface, then n − d +1=1. On the other hand, consider the case that X is a hypersurface in Pn for n ≥ 4. Then Pic(X)=1and α(X)= 1 . We therefore have that n−d+1 1 α(X)·τ∞(U)= ω 2 Leray CU∩N

for every measurable set U ⊆ X( Ê).

v. The Tamagawa measure.

6.21. Deﬁnition. The Tamagawa measure τH on the set X( É ) of adelic points on X is deﬁned to be the product measure τH := τν .

ν∈Val( É)

Sec. 6] peyre’s constant ii—other factors 57

6.22. Remark. X is projective. Therefore, one has that

X( É )= X( ν ) .

ν∈Val( É) 6.23. Lemma. The inﬁnite product

τν X( Éν ) ∈ É ν Val( ) is absolutely convergent. In particular, the infinite product measure τν ν∈Val( É) is well defined. Proof. Absolute convergence may not be destroyed by a finite set of factors. { }

Thus, assume that all the metrics . νp of the adelic metric . = . ν ν∈Val( É) are induced by a model X of X. Further, we may restrict the inﬁnite product to all prime numbers p such that X is smooth over p. Corollary 6.7 assures that X − # ( p) − 1 | Ip · ¶ τp X( Ép) =det 1 p Frobp Pic(X ) . ( ) É pdim X Further, smoothness over p implies that

Pic(X )Ip = Pic(X ) .

É É

We denote the eigenvalues of Frobp on Pic(X ) by λ1, ... ,λr.Aseverydivisor É on X is actually defined over a finite field extension, all these are roots of unity. p

Since Pic(X ) is a -module of ﬁnite rank, the characteristic polynomial of Frobp É 1 is monic with integral coeﬃcients. This shows, if λ is an eigenvalue, then λ = λ is an eigenvalue of Frobp,too. Clearly, one has −1 Ip −1 −1

det 1 − p Frobp | Pic(X ) =(1− λ1p ) · ... · (1 − λrp ) É −1 =1− (λ1 + ... + λr)p + E, where k k | E| < p−2 + ... + p−k 2 k k2 k3 < + + ... p2 p3 2k2 < p2 for p>2k.

In order to determine #X ( p), we use the Lefschetz trace formula in étale coho- 1 mology [SGA4 2 , Rapport, Théorème 3.2]. This yields X dim p

i i

− | X É #X ( p)= ( 1) tr Frobp H ( , l) ( ) ét p i=0 for every prime l = p.

58 conjectures on points of bounded height [Chap. II

Here, according to the Weil conjectures proven by P. Deligne [Del, Théorème (1.6)], i i/2 every eigenvalue of the Frobenius on H (X , Él) is of absolute value p . ét p The theorem on smooth base change implies a comparison theorem even for the unequal characteristic case [SGA4, Exp. XVI, Corollaire 2.5 and Exp. XV, Théorème 2.1]. We therefore know that

i ∼ i É

H (X , Él) = H (X , l) . É ét p ét

Accordingto[SGA4, Exp. XVI, Corollaire 1.6 and Exp. XI, Théorème 4.4], the

i É latter is isomorphic to the usual cohomology H (X( ), l).

É ≤ ∈ Æ Let C be a constant such that dim H (X( ), l) C for every i .Then

i ≤ dim H (X , Él) C ét p

for every i ∈ Æ, too. Further, Lemma 4.2 describes the ﬁrst and second cohomolo- 1 gies of a Fano variety more concretely. We have H (X , Él)=0while the ﬁrst ét p Chern class induces an isomorphism

∼ 2

⊗ É É

Pic(X ) l = H (X , l(1)) É É ét ∼ 2 = H (X , Él(1)) . ét p

This isomorphism is compatible with the operation of Frobp. Consequently, the 2 Frobenius eigenvalues on H (X , Él) are λ1p, ... ,λrp. ét p 2 dim X−1

Poincaré duality shows that H (X , Él)=0.Further,by[Del, (2.4)], ét p Poincaré duality is compatible with the Frobenius eigenvalues. Therefore, the eigen- 2 dim X−2 values of the Frobenius on H (X , Él) are ét p

dim X−1 dim X−1 λ1p , ... ,λrp .

For the number of p-rational points, this yields, according to ( ),

dim X dim X−1 dim X X ( p)=p +(λ1 + ... + λr)p + Dp , where 1 |D|

Altogether,

τp X( Ép) −1 −1 =(1− (λ1 + ... + λr)p + E) · (1 + (λ1 + ... + λr)p + D) 2 −2 −1 =1+E + D +(λ1 + ... + λr) p +(E − D)(λ1 + ... + λr)p + ED.

Sec. 7] peyre’s constant iii—the actual definition 59

For p>4C2, we certainly have 2k2 2C2 4C· C 4C |E| < < = 2 < . p2 p2 p3/2 ·p1/2 p3/2

Therefore, τp X( Ép) =1+X,where

8C C2 8C2 16C2 |X| < + + + p3/2 p2 p5/2 p3 2 2 2 8C + C + 8C + 16C < 2C 4C2 8C3 p3/2 2+ 17 C + 2 = 2 C . p3/2 Since 3/2 > 1, the inﬁnite product is absolutely convergent.

7. Peyre’s constant III—the actual deﬁnition i. Peyre’s deﬁnition.

7.1. Definition (E. Peyre, [Pe/T, Definition 2.4]). Peyre’s constant or Peyre’s Tamagawa type number is defined as

· · t Br − · τ(X):=α(X) β(X) lim (s 1) L(s, χPic(X )) τH X( É ) s→1 É for t = rk Pic(X).

⊆ É 7.2. Remark. Here, X( É ) X( ) denotes the part that is not af- fected by the Brauer–Manin obstruction. The Brauer–Manin obstruction will be

discussed in detail in Chapter IV. The precise deﬁnition of X( É ) is given in

Notation IV.2.5. According to Proposition IV.2.3.b.ii), X( É ) is a closed subset

of X( É ) and therefore measurable.

7.3. Thus, there is the following conjecture, which reﬁnes Conjecture 4.11.

Conjecture (Manin and Peyre). Let X be a smooth, projective variety over É. Assume that X is Fano. Denote by HK ∨ the anticanonical height function introduced in Deﬁnition 4.6. a) If dim X ≤ 2 or X ⊆ Pn is a complete intersection, then there exist a real number τ and a Zariski open subset X◦ ⊆ X such that

◦ rk Pic(X)−1

◦ ∨

{ ∈ É | }∼ NX ,HK ∨ (B)=# x X ( ) HK (x)

◦ rk Pic(X)−1

◦ ∨

{ ∈ É | }∼ NX ,HK ∨ (B)=# x X ( ) HK (x)

60 conjectures on points of bounded height [Chap. II

7.4. Remark (Some motivation). Morally, the conjecture of Manin in the reﬁned form due to Peyre states the following. The (conjectural) constant describing

the growth of the number of É-rational points on X is equal to a regularized product over the densities of the p-adic points on X together with the density of the real points. It is actually not very surprising that such a connection is expected. For instance, a low density of p-adic points on X for a particular prime p implies strong congruence

conditions for É-rational points. ii. Concluding remarks.

7.5. Remarks (Some more motivation). i) More precise results have been obtained by the classical circle method [Bir], cf. our remarks in the section below.

When applicable, it provides an asymptotic formula for the number of É-rational points on a variety X and an error term. The coeﬃcient of the main term is a product of p-adic densities together with a factor corresponding to the Archimedean valuation. Unlike the p-adic densities, the latter factor does not coincide in general

with E. Peyre’s factor τ∞ X( Ê) . Rather, it is an integral over the Leray measure, at least in the case that X is a hypersurface. · By Proposition 6.19, this may be rewritten in the form α(X) τ∞ X( Ê) .There- fore, one sees that the coeﬃcient of the main term, as provided by the circle method, is equal to

É · · Ê ·

τp X( p) α(X) τ∞ X( ) = α(X) τH X( É ) . p Here,

n k

{ ∈ X | ≡ } − 1 # y ( /p ) y a (mod p ) τp X( Ép) = 1 · lim . p n→∞ pn dim X − 1 The (1 p ) are convergence-generating factors. ii) It seems very natural to use (1 − 1 )t for t = rk Pic(X ) as the convergence- p É generating factors for the general case. Recall that, in the cases where the circle ∼

method is applicable, one always has Pic(X) = . É In the deﬁnition of Peyre’s constant, this is done. Indeed, the Gal( É/ )-

Ip ⊗ 

representation Pic(X ) contains the trivial representation t times. De- É note the complementary summand by P . Then the factors used in Deﬁnition 6.9 may be decomposed as 1 t −1 Ip −1 det 1 − p Frobp | Pic(X ) = 1 − · det 1 − p Frobp | P . É p −1 The factors det 1 − p Frobp | P form the Euler product for

1/L(s, χP ) at s =1. The value of the function L at 1 is introduced into Deﬁnition 7.1 only in order to cancel these factors out.

Sec. 7] peyre’s constant iii—the actual definition 61 iii) The definition of the factor α(X) is somehow more complicated and perhaps more mysterious than those of the other parts. Some motivation for the appearance of such a factor is given by the circle method. There is, however, another point that is at least as important. The definition of α(X) implies that the conjecture of Manin and Peyre is compatible with direct products of Fano varieties. iv) For a smooth cubic surface of the “first case” of Colliot-Thélène, Kanevsky, and Sansuc, the assertion of Theorem IV.6.4.c) implies that

Br 1

· É τ X( É ) = τ X( ) . H 3 H On the other hand, β(X)=3such that one might want to simplify Peyre’s formula to

· t − · ∗∗ τ(X):=α(X) lim (s 1) L(s, χPic(X )) τH X( É ) . ( ) s→1 É Clearly, this is also true in all cases when β(X)=1. Formula (∗∗) is, however, wrong in general. For example, when X is a smooth cubic surface, on which there is a Brauer–Manin obstruction to the Hasse principle, then we need τ(X)=0, and this is not provided by the simpliﬁed formula. Furthermore, in [Pe/T], E. Peyre and Y. Tschinkel reported strong numerical evidence for Peyre’s formula in cases similar to Example IV.5.36, below. I.e., if β(X)=3but the Brauer–Manin obstruction does not exclude any adelic

point on X, then there are three times more É-rational points on X than naively expected.

7.6. Remark. It is actually not all trivial that the conditionally convergent Euler product 1 − −1 | p det 1 p Frobp P really converges to L(1,χP ). The point is that the value L(1,χP ) is deﬁned by analytic continuation.

7.7. Remark. The idea behind Peyre’s constant had somehow been present a few years before E. Peyre gave his definition. The main term of the asymptotic formula provided by the circle method was used earlier, for example by D. R. Heath- Brown in his experimental investigations on cubic surfaces where weak approximation fails [H-B92a]. Actually, Heath-Brown’s point of view was a little different. He considered the − 1 t factors (1 p ) as convergence-generating factors growing out of the circle method. The resulting conditionally convergent infinite product was treated like a definition. The value of the function L at 1 appeared too, but only as part of a numerical method to speed up convergence. E. Peyre’s approach is most likely inspired by considerations like those in [H-B92a]. It has, however, the potential to work in much more generality.

62 conjectures on points of bounded height [Chap. II

7.8. Remark (Some evidence). i) D. R. Heath-Brown [H-B92a]aswellas E. Peyre and Y. Tschinkel [Pe/T] presented numerical evidence for Conjecture 7.3 for isolated examples of smooth cubic surfaces. Meanwhile, a bit more evidence of the same kind has been found [EJ6, EJ10]. In Chapter VI, we will report on our experiments providing numerical evidence e for the conjecture of Manin and Peyre in the case of the threefolds Va,b,givenby

axe = bye + ze + ve + we in P4 for e =3and 4. É ii) In certain particular cases, Manin’s conjecture has been proven. The goal of the next section is to give indications on the methods that have been used to do this.

8. The conjecture of Manin and Peyre—proven cases i. General remarks.

8.1. Generally speaking, Manin’s conjecture is wide open. But, nonetheless, there are quite a large number of particular cases, for which it is known. In fact, three methods have been established that may prove Manin’s conjecture for special kinds of varieties. And what makes the situation especially interesting is the fact that they are rather unequal in nature. Unfortunately, all three methods are rather involved. It would go beyond the scope of this book to present them in a way that could be considered complete. The plan for this section is nevertheless to give some indications and references, as well as a few details on one particular method, the so-called descent method. ii. The circle method.

8.2. The classical circle method belongs to the ﬁeld of analytic number theory. The basic idea is to apply the residue theorem in order to express the number ◦ { ∈ | ∨ } # x X ( É) HK (x)

Thus, for a smooth complete intersection X of multidegree d1, ... ,dn,thecir-

cle method provides an asymptotic formula for the number of É-rational points on X and an error term. Unfortunately, it is necessary to impose rather restrictive conditions on the dimension of the ambient projective space in order to ensure that the provable error term is smaller than the main term. The dimension has to be very large compared to d1,...,dn [Bir].

Sec. 8] the conjecture of manin and peyre—proven cases 63

When this is the case, this proves Manin’s conjecture in the refined form due to E. Peyre. In fact, the circle method was historically first and the definition of Peyre’s constant is motivated by its main term. Cf. Remarks 7.5, above.

8.3. Remarks. i) Contrary to what has been said, the circle method proves Conjecture 7.3, too, for linear subspaces of the projective space and for quadrics in arbitrary dimension. In these cases, sharp estimates are possible as there are no minor arcs occurring, in a certain sense. ii) It seems natural to expect that there should be a generalization to the multipro- jective case, and this is indeed the case. At least two situations have been worked out: that of a bilinear hypersurface in biprojective space [Rob], and that of a hypersurface in a very high-dimensional biprojective space [Schi13b, Theorem 1.1]. iii) For more information on the circle method, we would, ﬁrst of all, recommend that the interested reader study the lectures [Si63/64] of C. L. Siegel. Unfor- tunately, it seems that they are not available outside the Göttingen Mathemati- cal Institute. More recent presentations may be found in the article [Schm]of W. M. Schmidt or the book [Va] of R. C. Vaughan. iii. Universal torsors. The descent method.

8.4. Let X be a smooth, projective variety over É, and suppose that

Pic(X ) is ﬁnitely generated and free. For simplicity, let us also assume that all É

∼ r

invertible sheaves are already deﬁned over É. I.e., that Pic(X)=Pic(X ) = , É for some positive integer r. 1 ∼ It is well known [Mi, Example III.2.22] that, canonically, Hét(X, m) = Pic(X). Therefore, ∼ 1 ∗ Hom(Pic(X), Pic(X)) = Hét(X, Hom(Pic(X), m)) . ( )

The left-hand side contains the identity map as a distinguished element. Hence, the same must be true for the right-hand side. Furthermore, recall that, for G any 1 abelian group, the ﬁrst étale cohomology group Hét(X, G) classiﬁes G-torsors with respect to the étale topology [Mi, Corollary III.4.7].

∼ r 8.5. Deﬁnition. For NTX := Hom(Pic(X), m) = m the so-called Néron–Severi torus, the NTX -torsor over X corresponding to the distinguished 1 element of Hét(X, Hom(Pic(X), m)) is said to be the universal torsor over X.

8.6. Remarks. i) It is not hard to see that the universal torsor fulﬁlls a

t ∈ Æ universal property among all m-torsors over X,fort . t ii) Every m-torsor is in fact an X-scheme [Mi, Theorem III.4.3.a)]. In particular, the universal torsor is a scheme S, naturally equipped with a morphism S → X. ∼ r The ﬁbers over a K-rational point of X areisomorphictoNTX (K) = m(K). iii) One may as well consider torsors for the fppf- or fpqc-topology, as this is ac- r tually done in J. Milne’s textbook [Mi]. For m-torsors, this does not make any diﬀerence [Mi, Proposition III.4.9].

64 conjectures on points of bounded height [Chap. II

8.7. Lemma. Let X be a smooth, projective variety over É, and suppose ∼ r L L

that Pic(X)=Pic(X ) = , for some integer r. Choose a basis ( 1, ... , r) É of Pic(X). Then, for the universal torsor S over X, there is an isomorphism of X-schemes

∼ L \ ⊕ ⊕ L \ S = (V( 1) 0V(L1)) ... (V ( r) 0V (Lr)) .

Proof. The choice of the basis replaces the identity on Pic(X) by the homomor- r → L phism i: Pic(X), sending the k-th standard vector ek to k. On the other hand, the canonical isomorphism (∗) goes over into the isomorphism

r ∼ 1 r r

 Hom( , Pic(X)) = Hét(X, Hom( , m)) = Pic(X) , under which i is mapped to the r-tuple (L1,...,Lr). Finally, observe that replac-

ing an invertible sheaf by the corresponding m-torsor means nothing but taking the associated line bundle minus its zero section.

8.8. Remarks. i)InthecasethatPic(X) Pic(X ), the construction of É the universal torsor involves an additional step, where a Galois descent is to be per- formed. ∼ ii) The simplest special case one can think of is that Pic(X) = with a generator L that is very ample. Then the universal torsor is just the affine cone over i(X) without the cusp, for i: X → PN an embedding defined by L . iii) There is a slightly different description of the universal torsor, which has the potential to work in more generality. It is based on the Cox ring or universal coordinate ring of the variety X, which is constructed as follows.

For divisors D1, ... ,Dr ∈ Div(X) freely generating the Picard group, deﬁne the vector space 0 Cox(X):= H (X, OX (n1D1 + ... + nrDr)) . r (n1,... ,nr )∈

The isomorphisms

OX (n1D1 + ... + nrDr) ⊗ OX (m1D1 + ... + mrDr) ∼ = OX ((n1 + m1)D1 + ... +(nr + mr)Dr) induced by the tensor product of sections deﬁne a multiplication on Cox(X),trans- forming it into a ring that is equipped with a Pic(X)-grading. ∼ r The Néron–Severi torus NTX = m naturally operates on Cox(X).For ∈ 0 O · n1 nr s H (X, X (n1D1 + ... + nrDr)) and t =(t1, ... ,tr), put t s := t1 ... tr s. This extends to an action of NTX on the aﬃne scheme S := Spec Cox(X). The universal torsor is then the maximal open subset S ⊂ S, on which the operation of the Néron–Severi torus is faithful. Cox rings were originally introduced by D. A. Cox [Cox] in 1995. For further information about them, we advise the reader to study the survey article of A. Laface and M. Velasco [LV] or Chapter 2 of the Ph.D. thesis [Der] of Ulrich Derenthal.

Sec. 8] the conjecture of manin and peyre—proven cases 65

8.9. The fundamental idea of the method we are going to address here is the same as that of the “proof” for the statistical heuristic 1.4. There, instead of working on Pn, we preferred to consider An+1 \{0} and remarked that a positive proportion of the points has coordinates without common factors. The use of the universal torsor is the natural generalization of this approach to cases, where the Picard rank is possibly larger than one. Instead of working with the homogeneous coordinates corresponding to a specific projective embedding, one looks at the most general coordinates possible, which are described by the universal coordinate ring. For general X, the situation is no different from the case of Pn.ThepointsonX are in a direct relationship with the points on the universal torsor S, fulfilling certain extra conditions. These usually involve divisiblity or congruences. A bound for the anticanonical height should translate into a size condition. Thus, in order to prove Manin’s conjecture for a concretely given Fano variety X, one has to hope that the universal torsor S can be made sufficiently explicit. The problem is then reduced to counting the points on S fulfilling the extra conditions.

8.10. Remark. Universal torsors were introduced by J.-L. Colliot-Thélène and J.-J. Sansuc [CT/S76, CT/S87] originally in order to investigate the existence of rational points. The idea to use them for quantitative purposes is due to P. Salberger, who presented it ﬁrst at the Borel seminar at Bern in 1993. Cf. [Sab].

8.11. Remarks. i) At a ﬁrst glance, it seems to be quite a bizarre idea to work on the universal torsor S, which may be rather high-dimensional in comparison with X. But the point here is that a certain trade-oﬀ is going on. Its dimension may be higher, but, nevertheless, from an arithmetic point of view, S is somehow simpler than the original variety.

ii) In a certain sense, this observation can even be made quantitative. In fact, in É order to describe a É-rational point on X,the -rational point on S that is needed is usually of lower naive height. Cf. Proposition 8.18, where this eﬀect may be read oﬀ the formulas. For this reason, the use of the universal torsor is often called the descent method. iii) Nevertheless, it should be stressed that the determination of the universal torsor S just constitutes one ingredient of a possible proof of Manin’s conjecture for a particular variety. It may still be very hard or next to impossible to count the points of bounded height on S, to incorporate the list of extra conditions occurring, and to assemble all the details together in such a way that a complete argument arises.

8.12. Most publications in which the descent method is applied are concerned with isolated examples, not with large families. But the examples treated are often of big interest. They include extremal objects such as the cubic surface 4 with an E6-singularity [Der] and the surface, given in P by

− 2 x0x3 x1x4 = x0x1 + x1x3 + x2 =0, which is a quartic Del Pezzo surface with a singularity of type D4 [Der/T].

66 conjectures on points of bounded height [Chap. II

Many more cases have been investigated, most of them quite recently, and it would be impossible here even to list all of them. Let us just mention a number of results, without claiming completeness. i) Del Pezzo surfaces having special conﬁgurations of isolated singular points have been considered by R. de la Bretèche and T. D. Browning [dB/B07a, dB/B07b], R. de la Bretèche, T. D. Browning, and U. Derenthal [dB/B/D], T. D. Browning and U. Derenthal [B/D09a, B/D09b], P. Le Boudec [lB12a, lB12b, lB12c], and D. Loughran [Lou10, Lou12]. ii) As far as regular Del Pezzo surfaces are concerned, there is the work of R. de la Bretèche [dB02] on the degree ﬁve case, which was considered a big breakthrough at the time when it appeared. In the case of degree four, the conjecture of Manin and Peyre is, in general, still open. The particular situation when there exists a rational conic bundle structure has, however, recently been settled by R. de la Bretèche and T. D. Browning [dB/B11].

iii) Another interesting case is provided by the Chatelet surfaces over É arising as minimal proper models of the surfaces given by y2 + z2 = f(x),forf a separable

cubic polynomial that is split over É. For these, the conjecture of Manin and Peyre has been established by R. de la Bretèche and T. D. Browning, together with E. Peyre, in [dB/B/P]. There are two remarkable points in relation with these surfaces. First, they form a non-trivial family and, second, they are known to violate weak approximation, due to the Brauer–Manin obstruction. Up to now, proofs for Manin’s conjecture despite lack of weak approximation are very rare, the only other cases being varieties acted upon by an algebraic group. iv) In higher dimensions, after the invention of the universal torsor method by P. Salberger [Sab], the ﬁrst success were R. de la Bretèche’s new proofs [dB01] for results on toric varieties, originally due to V. V. Batyrev and Y. Tschinkel [Ba/T98]. Cf. Subsection v. Apart from these investigations, two impressive isolated results have been obtained. R. de la Bretèche treated the case of the Segre cubic threefold [dB07], while V. Blomer, J. Brüdern, and P. Salberger proved the conjecture of Manin and Peyre for a very particular cubic fourfold [B/B/S]. v) In the subsection below, we will give an impression on how the descent method works in practice, following D. R. Heath-Brown’s approach [H-B03]toproveaweak version of Manin’s conjecture for Cayley’s cubic surface.

8.13. Remarks. i) The examples mentioned in 8.12.i) actually have isolated singular points. This means that Conjecture 7.3 formally does not even apply to them. The minimal resolutions of singularities are isomorphic to P2 blown up in three to six points, not in general position. Such a surface is said to be weak Del Pezzo. The situation is similar for the examples given in 8.12.iv). These are actually almost Fano varieties inthesenseofE.Peyre[Pe03, Déﬁnition 3.1], as are the weak Del Pezzo surfaces.

Sec. 8] the conjecture of manin and peyre—proven cases 67

There is, in fact, a slightly more general version of the conjecture of Manin and Peyre, also due to E. Peyre [Pe03, Formule empirique 5.1], that applies to a large class of almost Fano varieties. This includes all the cases mentioned above. ii) We formulated Manin’s conjecture only for smooth Fano varieties, which is, as we strongly believe, in accordance with Yu. I. Manin’s original intentions. One might even tend to think that a generalization to singular varieties is unnecessary or at least irrelevant. Indeed, the main reason why such a generalization is being considered is a very pragmatic one. Singularities often tend to make the asymptotics of the number of rational points of bounded height a bit easier to understand. For example, Manin’s conjecture could not yet be established for a single smooth cubic surface. On the other hand, among the examples given in 8.12, there are a number of singular cubic surfaces occurring. Further, D. R. Heath-Brown obtained an interesting partial result for the Cayley cubic, on which we will report in the next subsection. But, perhaps, the eﬀect becomes most obvious when one takes a look at the cubic 3 surface given by x0 = x1x2x3, which is rather singular, having three binodes. For this surface, the conjecture of Manin and Peyre is proven and experts would probably say that it is easy to do this. Moreover, unusually sharp estimates have been obtained for the error term; cf. paragraph 8.32, below. In addition, this is one the relatively few cases in which more than one method applies for the proof of Manin’s conjecture.

8.14. To obtain more information about this particular method, the reader is advised to study the articles [Der/T]ofU.DerenthalandY.Tschinkeland[Bro] of T. D. Browning, which are excellent surveys. iv. A lower bound for the number of rational points on Cayley’s cubic surface.

8.15. Cayley’s cubic surface X is given in P3 by the equation É

x0x1x2 + x0x1x3 + x0x2x3 + x1x2x3 =0. (†)

It has exactly four singular points, at (1:0:0:0), (0:1:0:0), (0:0:1:0),and (0:0:0:1), which are conical ordinary double points. Further, X contains exactly

nine lines, all of which are deﬁned over É. Six of them are of the type xi = xj =0, the three others are given by x0 + x1 = x2 + x3 =0and the analogous equations. The minimal resolution of singularities of X is isomorphic to P2 blown up in six points, not in general position. This is a weak Del Pezzo surface of Picard rank seven. The generalized version of Manin’s conjecture predicts, for the number

of É-rational points on X of naive height bounded by B, a growth of the type τBlog6B. It is not yet known whether this prediction is correct. However, in [H-B03], D. R. Heath-Brown proved the following result.

68 conjectures on points of bounded height [Chap. II

8.16. Theorem (D. R. Heath-Brown). Let X be Cayley’s cubic surface, and let X◦ be its open subset obtained by removing the nine lines. Then there is a ∈ positive constant c Ê such that

6 ◦ cB log B

8.17. Remarks. i) D. R. Heath-Brown also proves an upper bound of the form CB log6B. We will only explain here how to obtain the lower bound. For the upper bound, similar techniques are used, but the details are more involved. ii) It seems that, generally, lower bounds are somewhat easier to obtain than upper bounds. For example, in [Sl/SD], J. B. Slater and Sir Peter Swinnerton-Dyer −

proved a lower bound of the form cB logrk Pic(X) 1B,forX an arbitrary smooth É cubic surface over É containing two skew -rational lines. No analogue of this result is known when upper bounds are concerned. Of course, the result of Slater and Swinnerton-Dyer does not apply here, as X has singular points.

8.18. Proposition. Let X be Cayley’s cubic surface, let X◦ be its open ◦ subset obtained by removing the nine lines, and let (t0 : ... : t3) ∈ X ( É) be an arbitrary point. Suppose that t0, ... ,t3 are normalized to be integers without common factor. a) Then there exist positive integers z01, z02, z03, z12, z13, z23 and non-zero integers y0,y1,y2,y3, fulﬁlling the conditions

gcd(zij,zkl)=1 for (i, j) =( k, l) ,

gcd(yi,yj )=1 for i = j, (‡)

gcd(yi,zij)=1 for i = j, and

z12z13z23y0 + z02z03z23y1 + z01z03z13y2 + z01z02z12y3 =0, (§) but

z12z13z23y0 + z02z03z23y1 =0 ,

z12z13z23y0 + z01z03z13y2 =0 , (¶)

z12z13z23y0 + z01z02z12y3 =0 , such that

t0 = z01z02z03y1y2y3 ,

t1 = z01z12z13y0y2y3 , ( )

t2 = z02z12z23y0y1y3 ,

t3 = z03z13z23y0y1y2 .

Sec. 8] the conjecture of manin and peyre—proven cases 69

Thereby, the coordinates t0,...,t3 uniquely determine z01,z02,z03,z12,z13,z23,as well as y0, y1, y2,andy3. b) On the other hand, for every choice of z01, ... ,z23, y0, ... ,y3, fulﬁlling the conditions (‡), (§),and(¶), the formulas ( ) yield a quadruple of integers, having ◦ no factor in common and deﬁning a point (t0 : ... : t3) ∈ X ( É). Proof. a) We will prove this part in several steps. First step. Common factors of three coordinates.

Put ηi := gcd(tj ,tk,tl), for every i and { j, k, l } = {0, 1, 2, 3}\{i}. Then, as not all four coordinates may have a factor in common, we have gcd(ηi,ηj )=1,fori = j. This, in turn, shows that we may write ti = ηj ηkηl·zi,for{ j, k, l } = {0, 1, 2, 3}\{i} and suitable integers zi. These formulas imply, in particular, that no three of the zi may have a common factor. Further, gcd(ηi,ti)=1, which yields gcd(ηi,zi)=1. Finally, the equation of the surface goes over into

z1z2z3η0 + z0z2z3η1 + z0z1z3η2 + z0z1z2η3 =0.

Second step. Common factors of two zi’s.

Let us put zij := gcd(zi,zj ),fori = j. We will not distinguish between zji and zij. Then, as no three of the zi have a common factor, we clearly have gcd(zij,zkl)=1, for {i, j} = {k, l}. This shows that we may write zi = zijzikzil ·wi, for suitable integers wi and { j, k, l } = {0, 1, 2, 3}\{i}. By construction, the wi must be mutually coprime. The fact that gcd(ηi,zi)=1implies gcd(ηi,zij)=1, for any i = j. The equation of the surface takes the form

z12z13z23w1w2w3η0 + z02z03z23w0w2w3η1

+ ···+ z01z03z13w0w1w3η2 + z01z02z12w0w1w2η3 =0.

Third step. wi = ±1 for i =0, ... ,3.

Let us show that w0 = ±1, the other facts being analogous. The equation just obtained yields w0|z12z13z23w1w2w3η0, which, as the wi are mutually coprime, implies w0|z12z13z23η0. Next, we realize that gcd(wi,zjk)=1, whenever i = j and i = k. Indeed, a common factor would divide zi,zj and zk. Thus, we see that even w0|η0.

We put yi := wj wkwl ·ηi,for{ j, k, l } = {0, 1, 2, 3}\{i}. Then the equation of the surface goes over into (§). The property of the point (t0 : ... : t3) not to lie on any of the lines of type x0 + x1 = x2 + x3 =0is still encoded by the requirement that no sum of two summands already vanishes. The coprimality conditions (‡) are implied by the corresponding ones for η0, ... ,η3 and z01, ... ,z23, established during the ﬁrst and second steps.

70 conjectures on points of bounded height [Chap. II

Finally, by construction, we have

yj yk yl ti = zi ·ηj ηkηl = zijzikzilwi ·ηj ηkηl = zijzikzilwi · · · wiwkwl wiwj wl wiwj wk yj ykyl = ... = zijzikzil 2 = zijzikzilyj ykyl , (wiwj wkwl) as asserted in ( ). Fifth step. Uniqueness. We ﬁrst claim that the formulas ( ) together with the coprimality conditions (‡) imply that |yi| =gcd(tj ,tk,tl),for{ j, k, l } = {0, 1, 2, 3}\{i}.Toprovethis,letus verify |y0| =gcd(t1,t2,t3), the other assertions being analogous.

First of all, we see from ( )thatt1, t2,andt3 are divisible by y0. Hence, it suﬃces to verify that gcd(z01z12z13y2y3,z02z12z23y1y3,z03z13z23y1y2)=1.Inordertodo this, recall that the zij are mutually coprime. Consequently,

gcd(z01z12z13,z02z12z23,z03z13z23)=1.

Thus, any common prime factor p must necessarily divide one of the factors yi. Suppose without restriction that p|y1. Then, by the assumptions, p cannot divide any of the factors z01, z12, z13, y2,andy3, which is a contradiction.

The fact that |yi| =gcd(tj,tk,tl) shows that the coordinates t0, ... ,t3 determine the integers y0, ... ,y3, at least up to sign. But, as the zij are positive and the

4 4

→ group homomorphism s:(/2 ) ( /2 ) , given by taking all possible sums of three coordinates, is injective, we see that the signs are ﬁxed by t0,...,t3,too.

Consequently, the coordinates t0, ... ,t3 determine, as well, the four expressions z01z02z03, z01z12z13, z02z12z23,andz03z13z23. We may compute

z01 := gcd(z01z02z03,z01z12z13) , and analogously for the other zij . Uniqueness is proven. b) In the final step of the proof of a), we showed that the coprimality conditions (‡)imply,fort0,...,t3 defined by the formulas ( ), that |y0| =gcd(t1,t2,t3). Since y0 is relatively prime to z01, z02, z03, y1, y2,andy3, this immediately yields gcd(t0,...,t3)=1. Further, plugging the formulas ( ) into the left-hand side of (†), one finds a reducible polynomial, splitting into the left-hand side of (§) and the factor 2 z01 · ... · z23(y0 · ... · y3) . As the equality (§) is assumed, the equation (†)

must be true. We have (t0 : ... : t3) ∈ X( É). Further, no sum of two summands may vanish already, as the same is true for (§). ◦

Consequently, we see that, actually, (t0 : ... : t3) ∈ X ( É). This completes the proof.

◦

8.19. Remarks. i) Clearly, the point (t0 : ... : t3) ∈ X ( É) determines z01,...,z23 uniquely and y0,...,y3 uniquely up to a common sign. ii) The nine-dimensional auxiliary variety, given in A10 by the equation (§), is actually not yet the universal torsor of X, but only some sort of approximation.

Sec. 8] the conjecture of manin and peyre—proven cases 71

◦

It certainly yields a nice parametrization of X ( É) that will turn out to be suﬃcient for our purposes. Cf. [H-B03, Section 2] or [Der, Example 6.7] for equations of the actual universal torsor of X.

8.20. Notation. i) For B>0 and δ>0 arbitrary, we deﬁne a subset of 10

10 | MB,δ := { (z01,...,z23,y0,...,y3) ∈ δ z01,...,z23 > 0,y0,...,y3 =0 ,P:= z01 · ... · z23

6 ii) Further, for B,δ > 0 and positive z01,...,z23 ∈ , we put

{ ∈ | ∈ NB,δ;z01,... ,z23 := # (y0,...,y3) (z01,...,z23,y0,...,y3) MB,δ , conditions (§)and(‡) are fulﬁlled } .

8.21. Lemma. a) The number of 10-tuples (z01, ... ,z23,y0, ... ,y3) ∈ ◦ MB,δ, fulﬁlling the conditions (§) and (‡), that yield a point on X \ X 2 + 20 δ is O(B 3 3 ). b) For some B, δ>0, suppose that

(z01,...,z23,y0,...,y3) ∈ MB,δ .

Further, assume that equation (§) and the coprimality conditions (‡) are fulﬁlled,

but the conditions (¶) are possibly not. Then, for the point (t0 : ... : t3) ∈ X( É), corresponding to (z01,...,z23,y0,...,y3), one has Hnaive(t0 : ... : t3)

δ δ Proof. a) First of all, z01 · ...· z23

z12z13z23y0 + z02z03z23y1 = z01z03z13y2 + z01z02z12y3 =0 has to be satisﬁed. This means that, as soon as (z01, ... ,z23) is ﬁxed, two of 2 2 + 2 δ the yi determine the others and we do not have more than O((BP) 3 )=O(B 3 3 ) possibilities for them. b) We have BP |t0| = z01z02z03|y1||y2||y3|

≤ 1 8.22. Proposition. Let 0 <δ 84 be arbitrary. Then there exist constants cδ > 0 and B0,δ > 0 such that the following assertion is true.

72 conjectures on points of bounded height [Chap. II

6 Let B>B0,δ. Then, for any sextuple (z01, ... ,z23) ∈ of mutually coprime, δ positive integers that satisfy the inequality P = z01 · ... · z23 0, mutually coprime, · · δ P :=z01 ... z23

8.24. Lemma. For N →∞, N ϕ(n) d (n) log6N, 6 n2 n≥1; 2,3 n n square-free where d6(n) denotes the number of representations n = n1 · ...· n6 as a product of positive integers. →∞ M ϕ(n) Proof. It is well known that, for M , n=1 n2 log M [Apo, Exercise 3.6, ϕ(2n) ≤ 1 ϕ(n) ϕ(3n) ≤ 1 ϕ(n) cf. Theorem 3.7]. Further, (2n)2 2 n2 and (3n)2 3 n2 immediately show that M ϕ(n) 5 M ϕ(n) M ϕ(n) 1 M ϕ(n) ≤ and, thus, ≥ log M n2 6 n2 n2 6 n2 n=1 n=1 n=1 n=1 2|n or 3|n 2n,3n too. Consequently, M · · ϕ(n1) ... ϕ(n6) 6 2 log M. (n1 · ... · n6) n1, ... ,n6=1 2,3 n1, ... ,n6 Now let p ≥ 5 be any prime. The inequality · · · · ϕ(pn1)ϕ(pn2)ϕ(n3) ... ϕ(n6) ≤ 1 ϕ(n1) ... ϕ(n6) 2 2 2 (pn1 · pn2 · n3 · ... · n6) p (n1 · ... · n6) shows M · · M · · ϕ(n1) ... ϕ(n6) ≤ 15 ϕ(n1) ... ϕ(n6) 2 2 2 , (n1 · ... · n6) p (n1 · ... · n6) n1, ... ,n6=1 n1, ... ,n6=1 2,3 n1, ... ,n6 2,3 n1, ... ,n6 p divides at least two and therefore M · · M · · ϕ(n1) ... ϕ(n6) ≥ − 15 · ϕ(n1) ... ϕ(n6) 2 1 2 2 . (n1 · ... · n6) p (n1 · ... · n6) n1, ... ,n6=1 p≥5 n1, ... ,n6=1 mutually coprime prime 2,3 n1, ... ,n6 2,3 n1, ... ,n6

Sec. 8] the conjecture of manin and peyre—proven cases 73

The right-hand side is still log6M,forM →∞, since the inﬁnite product 1 − 15 6 p≥5(1 p2 ) converges to a non-zero limit. Putting M := N , we obtain the claim.

8.25. Remark. The main tool, which is used in the proof of the inequality M ϕ(n) n=1 n2 log M, is Perron’s formula [Apo, Theorem 11.18 or Brü, Satz 1.4.4]. In fact, Perron’s formula is strong enough to yield the whole of Lemma 8.24. For this, one has to study the Dirichlet series N ϕ(n) F (s):= d (n) n−s , 6 n2 n≥1; 2,3 n n square-free 6(p−1) −s which has the Euler product expansion p∈È [1 + 2 p ]. p=2 ,3 p After a few elementary estimates, the argument ﬁnally comes down to showing xs that ress=0 F (s) s is a polynomial of degree six in log x with a positive leading co- eﬃcient. This, however, is a standard calculation for F (s) replaced by ζ6(s +1); cf. [Brü, formula (4.20)]. Further, the quotient F (s)/ζ6(s +1)is easily seen to be holomorphic for Re(s) > −1 and strictly positive at s =0.

8.26. Proof of Proposition 8.22. First step. Let A0, ... ,A3 and h0, ... ,h3 be integers, diﬀerent from zero. De-

note by ZA0,... ,A3,h0,... ,h3 (D) the number of integral solutions of the equation

A0y0 + ... + A3y3 =0,

lying in the hypercuboid |yi|

gcd(A0h0,... ,A3h3) 3 2 2 2 2 2 ZA ,... ,A ,h ,... ,h (D)=C D + O(A h · ... · A h ·D ) , 0 3 0 3 A0h0·... ·A3h3 0 0 3 3 for D →∞. Here, C is a positive constant, which could easily be made explicit. Further, the constant implicitly contained in the error term is independent of A0,...,A3,aswellash0,...,h3.

To prove this claim, we ﬁrst observe that the integer ZA0,... ,A3,h0,... ,h3 (D) equals the number of solutions of w0 + ... + w3 =0such that |wi|

A0h0|w0,A1h1|w1,A2h2|w2,A3h3|(w0 + w1 + w2) ,

as well as |w0|, |w1|, |w2|, |w0 + w1 + w2|

us with a lattice Λ, the index of which in is equal to

lcm(A3h3,gcd(A0h0,A1h1,A2h2)) A0h0·... ·A3h3 A0h0 · A1h1 · A2h2 · = . gcd(A0h0,A1h1,A2h2) gcd(A0h0,... ,A3h3)

By construction, ZA0,... ,A3,h0,... ,h3 (D) is the number of lattice points contained in 3 | | | | | | | | the polyhedron in Ê ,givenby w0 , w1 , w2 , w0 + w1 + w2

74 conjectures on points of bounded height [Chap. II

gcd(A0h0,... ,A3h3) 3 that ZA ,... ,A ,h ,... ,h (D) is asymptotically equal to C D ,forC 0 3 0 3 A0h0·... ·A3h3 the volume of

3 || | | | | | | | } P := { (w0,w1,w2) ∈ Ê w0 < 1, w1 < 1, w2 < 1, w0 + w1 + w2 < 1 .

There is an aﬃne transformation normalizing the lattice to and it is a well-known fact that the error of the asymptotic formula obtained is of the order of the boundary of the polyhedron appearing as the image of P under such a transformation. Cf. [Mc, Chapter 6, Proof of Lemma 2]. We will estimate the boundary size of a convex polyhedron very simply by the square 3 → of its diameter. Thus, let M : Λ be an isomorphism of lattices. The error −1 2 term is then O(( M max · D) ),for . max the maximal absolute value of the coeﬃcients of a matrix. 3

As |det Λ| =# /Λ, there are the three linearly independent lattice vectors (|det Λ|, 0, 0), (0, |det Λ|, 0),and(|det Λ|, 0, 0). Consequently, there is a lattice basis for Λ, consisting of three vectors from the box [0, |det Λ|]3. In particular, we may choose the 3 × 3-matrix M in such a way that all its nine coeﬃcients are in the interval [0, |det Λ|]. In view of the description of M −1 via the adjugate matrix of M, −1 this implies M max ≤|det Λ|. 2 2· · 2 2 2 2 A0h0 ... A3h3 2 Thus, for the error, there is the bound O(det Λ·D )=O( 2 ·D ), gcd(A0h0,... ,A3h3) and this is enough to imply the claim. Second step. Small primes. The main term.

We will use the notation Bi := zijzikzil,for{ j, k, l } = {0, 1, 2, 3}\{i} and i arbitrary. Similarly, we put Ai := P/Bi.

Then equation (§)issimplyA0y0 + ... + A3y3 =0. Further, the requirement that 1 (z01,...,z23,y0,...,y3) ∈ MB,δ means nothing but |yi| < (BP) 3 /Ai.

The coprimality conditions (‡)gooverintotherequirementsthatgcd(yi,Bi)=1, for i =0, ... ,3, and that the yi be mutually coprime. We will√ slightly relax the latter condition, asking the yi only to have no prime factors p ≤ log B in common, as well as none of the primes dividing P . The corresponding number of solutions will be denoted by N = N . B,δ;z01,... ,z23 We will show in the ﬁfth step that the error

| − | NB,δ;z01,... ,z23 NB,δ;z01,... ,z23 caused by this relaxation is of negligible size. In order to estimate N , we start by recalling that the number of solutions of (§)that are contained in the set MB,δ and fulﬁll di|yi as well as dij|yi,yj ,for0 ≤ i, j ≤ 3, 1 3 is exactly the quantity ZA0,... ,A3,h0,... ,h3 ((BP) ), investigated in the ﬁrst step. Here, we put hi = lcm(di,dij,dik,dil) for { j, k, l } = {0, 1, 2, 3}\{i}. Therefore, a standard application of the inclusion-exclusion principle shows that 1 · · · · 3 ∗∗ N = μ(d0) ... μ(d3) μ(d01) ... μ(d23) ZA0,... ,A3,h0,... ,h3 ((BP) ) , ( )

di|Bi dij |Q

Sec. 8] the conjecture of manin and peyre—proven cases 75 · √ for μ, as usual, the Möbius function and Q := P p< log B p. According to the result of the ﬁrst step, we replace this sum by gcd(A0h0,... ,A3h3) N := C·BP · μ(d0) · ... · μ(d3) μ(d01) · ... · μ(d23) . A0h0·... ·A3h3 di|Bi dij |Q ϕ(P ) We will prove next that indeed N B P 2 . In the fourth step, we will show that the error |N − N | is of smaller order. Third step. Evaluation of the sum. (p) f For any prime number p and i =0, ... ,3,wewriteAi := p if p appears in the prime decomposition of Ai to the exact exponent f. The multiplicativity of the Möbius function then ensures that

(p) (p) gcd(A h ,... ,A h ) · · · · · · 0 0 3 3 N = C BP μ(d0) ... μ(d3) μ(d01) ... μ(d23) (p) (p) . | | A h0·... ·A h3 p prime di Bi dij Q 0 3 p|Q d ∈{1,p} ∈{ } i dij 1,p Concerning the values of the factors that occur here, there are two cases.

Case 1. p z01,...,z23.

Then p Bi and p Ai, for any i. The factor corresponding to the prime p turns out to be 1 min(mi)−(m0+... +m3) gcd(h0,... ,h3) e01+... +e23 μ(d01) · ...· μ(d23) = (−1) p i , h0·... ·h3 dij ∈{1,p} e01,... ,e23=0 − 6 5 for mi := maxj (eij). This sum of 64 terms evaluates to 1 p2 + p3 .

Case 2. p divides exactly one of the integers z01,...,z23.

We assume without restriction that p|z01. Further, we denote by e ≥ 1 the exact exponent of p in the prime decomposition of z01.Thenp|B0,B1,A2,A3,eachtothe exact exponent e, while the other Ai and Bi are relatively prime to p.Thefactor corresponding to p is therefore

e e · · gcd(h0,h1,p h2,p h3) μ(d0)μ(d1)μ(d01) ... μ(d23) 2e p h0h1h2h3 d0,d1,dij ∈{1,p} 1 · · gcd(h0,h1) = 2e μ(d0)μ(d1)μ(d01) ... μ(d23) p h0h1h2h3 d0,d1,dij ∈{1,p} 1 − 1 − e0+e1+e01+... +e23 min(m0,m1) (m0+... +m3) = p2e ( 1) p , e0,e1,e01,... ,e23=0 where mi := max {ei,eij | j = i} for i =0, 1 and mi := maxj (eij) for i =2, 3. − 1 − 1 1 This sum of 256 terms evaluates to 1 p p2 + p3 . 1− 1 − 1 + 1 − 6 5 p p2 p3 Next, we observe that the inﬁnite product p(1 p2 + p3 ) 1− 1 converges p 1 to a non-zero limit ι. Further, the product over all the factors p2e that occur 1 is exactly P 2 , such that we see that 1 B ϕ(P ) N >C·BP · ι (1 − 1 )=ιC· (1 − 1 )=ιC·B . P 2 p P p P 2 p p This is precisely the lower bound desired.

76 conjectures on points of bounded height [Chap. II

Fourth step. The error of the lattice point estimates. 1 3 According to the ﬁrst step, replacing the term ZA0,... ,A3,h0,... ,h3 ((BP) ) by gcd(A0h0,... ,A3h3) C·BP · · · in an individual summand of the sum (∗∗) causes an error A02h0 ... A3h3 3 · 2 2 · · 2 2 of O((BP) A0h0 ... A3h3).

Further, by construction, we have hi|BiQ and, therefore,

2 · · 2| 2 · · 2 8 4 8 h0 ... h3 B0 ... B3 Q = P Q .

2 · · 2 4 On the other hand, A0 ... A3 = P . Altogether,

2 2 2 26 3 · 2 2 · · 2 2 ≤ 3 8 8 ≤ 3 + 3 δ 8 (BP) A0h0 ... A3h3 (BP) P Q B Q .

Finally, the prime number theorem√ [Apo, Chapter 4, formula (1) or Brü, Satz 1.1.4] implies that Q P exp(O( log B)) Bδ·Bδ = B2δ. Thus, the error per individual 2 + 74 δ summand is O(B 3 3 ). δ 2δ Furthermore, we have Bi ≤ P0 [Brü, Satz 1.2.1]. Hence, the number of divisors di and dij, respectively, may be bounded by δ δ B 10 and the number of all 10-tuples (d0,...,d3,d01,...,d23) by B . 2 + 77 δ Consequently, for the error term in total, we find an estimate of O(B 3 3 ). To conclude, we have to show that this is asymptotically smaller than the main ϕ(P ) term B P 2 . In order to see this, recall that, by [Apo, Theorem 13.14], ϕ(P ) · 1 ≥ · 1 1−2δ B P 2 B P log log P B Bδ log log(Bδ) B , ≤ 1 2 77 − and observe that δ 84 is sufficient for 3 + 3 δ<1 2δ. Fifth step. The other error term, concerning large primes. We still have to discount the effect of the relaxation undertaken in the second step. For this, let us estimate from above the number of solutions (y0, ... ,y3) of the | | equation A0y0 + ... + A3y3 =0, satisfying the size conditions yi √ log B. We will use that gcd(p, P )=1. On the other hand, the coprimality conditions may be ignored here. It will turn out that the upper bound obtained is still sufficiently small.

Without restriction, assume that p divides y0 and y1. Then the number of possi- D bilities for y0 is obviously O( ).Further,z01|A2 and z01|A3 such that we have A0p the congruence condition A0y0 + A1y1 ≡ 0(mod z01).Asp is relatively prime to D z01, this shows that there are not more than O( +1)options for y1,assoon A1pz01 as y0 has been chosen.

Continuing in this way, for every admissible pair (y0,y1), one ends up with an equation of the form A2y2 +A3y3 = c. Here, the two coeﬃcients are divisible by z01, A2 A3 A3 as is c.But and are coprime such that the equation ﬁxes (y2 mod ). z01 z01 z01 The total number of solutions for an individual large prime p is therefore O D ( D +1)(Dz01 +1) . A0p A1pz01 A2A3

Sec. 8] the conjecture of manin and peyre—proven cases 77

Our next goal is to simplify this estimate. This is easy for the third factor. Indeed, 1 1 1 2 2δ Dz01 −2δ Dz01 ≥ D =(BP) 3 ≥ B 3 and A2A3 ≤ P B3 > 1, A2A3 as δ<1 .Consequently, Dz01 +1 Dz01 . 6 A2A3 A2A3 1 1 D1/2 D Further, we have A1z01 ≤ P ≤ (BP) 6 = D 2 , i.e., 1 ≤ and, ﬁnally, p< , A1z01 A1 hence 1 < D . Together, these inequalities show that A1p D1/2 · D1/2 D ≤ D 1 < 1/2 = 3/2 1/2 , A1z01 1/2 1/2 A1p z01 A1 p A1 p z01 D D and, hence, +1 1/2 . A1pz01 A1p z01

Therefore, the number of solutions of (§), satisfying |yi|

3 D · D · Dz01 D BP B 1/2 = 3/2 = 2 3/2 = 3/2 . A0p A1p z01 A2A3 A0·... ·A3p P p Pp But 1/4 B < 2B < 2B = 2δ B , √ Pp3/2 P log1/4B P log1/4(P 1/δ) P log1/4P p> log B B ϕ(P ) which is of negligible size in comparison with P log log P B P 2 [Apo,Theorem 13.14]. This completes the proof.

8.27. Remark. It might seem surprising that such a vast amount of can- cellation occurs when evaluating the 64-term and 256-term sums in the third step. The point is that the sums may be interpreted in a diﬀerent way, which at least yields an explanation for the formulas obtained. The 64-term sum actually counts, again by the inclusion-exclusion principle, the 4 number of solutions of x0 + x1 + x2 + x3 =0in p such that not more than one 1 summand vanishes, multiplied by a factor p3 . 1 Similarly, the 256-term sum counts, up to a factor of p6 , the number of solutions

2 4 2 ∗

∈ of x0 + x1 + px2 + px3 =0in ( /p ) such that x0,x1 ( /p ) and not both x2 and x3 are divisible by p. But both these numbers are easier to be determined directly, not making use of the inclusion-exclusion principle, and this explains why the resulting formulas are that simple. v. Varieties with many symmetries.

8.28. There is an entirely diﬀerent method to approach the conjecture of Manin, which was invented by J. Franke, Yu. I. Manin, and Y. Tschinkel [F/M/T]. The original situation was that of a generalized ﬂag variety, i.e., of a quotient G/P , where G is a linear algebraic group and P ⊂ G a parabolic subgroup. Gener- ally speaking, the method is restricted to varieties with many symmetries. The fundamental idea is consider the height zeta function −s ZK ∨,U (s):= HK ∨ (x) .

x∈U( É)

78 conjectures on points of bounded height [Chap. II

In this deﬁnition, U ⊆ X may be an arbitrary Zariski open subset. It turns out, however, that the height zeta function may be investigated particularly well when U = T is a commutative group scheme. Then the Poisson summation formula from Fourier analysis, cf. [Ge, §9], yields a representation by an integral as follows, ZK ∨,U (s)= H(χ) dχ .

∨

 É

(T ( É )/T ( )) −s Here, H is the Fourier transform of HK ∨ (x) , extended to all adelic points, which is given by the formula −s

H(χ):= HK ∨ (x) χ(x) dx . 

T ( É )

∨

In these formulas, (T ( É )/T ( )) denotes the group of all continuous characters

É É on T ( É ) that are trivial on T ( ), dx is a Haar measure on T ( ),anddχ denotes the measure dual to dx. This representation may now be used in order to establish analytic properties of the height zeta function, in a way vaguely analogous to classical investigations for the Riemann zeta function, cf. [Ne, §VII.1]. For example, it is proven [F/M/T, Theorem 5] that the height zeta function has a meromorphic continuation to the whole complex plane, a pole of order rk Pic(X) at s =1, and no other poles in the half-plane Re(s) ≥ 1. The leading coeﬃcient of the Laurent series at s =1is equal to (rk Pic(X) − 1)! τ(X) , cf. [Ba/T98, Theorem 7.3]. A Tauberian argument is ﬁnally used in order to prove exactly the desired asymptotics.

8.29. Remarks. i) The method of Franke, Manin, and Tschinkel is naturally restricted to a particular class of varieties. But, similarly to the method ∼ discussed before, there is no limitation to the case that Pic(X) = . ii) Besides for generalized flag varieties [F/M/T], Conjecture 7.3 could be con- firmed for projective, smooth toric varieties. I.e., for equivariant compactifications r of the group m [Ba/T98]. Furthermore, M. Strauch and Y. Tschinkel treated the case of a toric fibration over a generalized flag variety [S/T]. Another interesting situation is that of an equivariant compactification of the vec- r tor group a. In this case, Manin’s conjecture in the refined form due to E. Peyre was established by A. Chambert-Loir and Y. Tschinkel [C-L/T]. Note here that r these are not isolated examples. An equivariant compactification of a allows deformations. iii) To obtain more detailed information, particularly on this method, we suggest that the reader study the original works of Franke, Manin, and Tschinkel [F/M/T] or Batyrev and Tschinkel [Ba/T98], and take a look into the survey article [Pe02] of E. Peyre, which is far from being outdated slightly more than ten years after it was written.

Sec. 8] the conjecture of manin and peyre—proven cases 79 vi. Higher order terms.

8.30. In some cases, the proof for Manin’s conjecture actually yields

◦ for NX ,Hnaive (B) a more precise asymptotic formula than the one in Conjecture 7.3, predicted by Manin and Peyre. This is certainly an exceptional situation, even among the particular cases, for which Manin’s conjecture is proven. To present a speciﬁc example, let us consider the cubic surface X that is given in P3 by the equation x3 = x x x . É 0 1 2 3

8.31. Remarks. i) The cubic surface X has three É-rational binodes. Thus, once again, Conjecture 7.3 formally does not apply to it. Manin’s conjecture in the reﬁned version due to Peyre is true for the weak Del Pezzo surface obtained by minimally resolving the singularities of X. ii) On X, there are exactly three lines. This is classically known for cubic surfaces with this particular conﬁguration of singularities [Dol, Table 9.1]. The lines are

contained in the plane, defined by x0 =0. As they are defined over É, for Manin’s conjecture, they have to be excluded from the count. iii) Thus, the problem is reduced to the affine surface, given by the equation x1x2x3 =1. It seems obvious that this equation has been studied numerous times during the history of mathematics.

8.32. Manin’s conjecture for the variety X was proven several times and by various people. It is noticeable that the torsor method has been applied as well as the height zeta function method. And both methods provide us with a description

◦ of the asymptotics of NX ,Hnaive (B) that is more precise than that predicted by the general conjecture of Manin and Peyre. The formulas known today involve several terms of higher order. Not claiming that the list is complete, let us mention the results below. First of all, X is a singular toric variety such that the work of Batyrev and Tschinkel [Ba/T98] applies. Further, D. R. Heath-Brown and B. Z. Moroz [H/M] gave a nice and rather elementary proof for the main term. The same result was obtained independently by È. Fouvry [Fou]. From a formal point of view, these elementary proofs are certainly to be considered as being applications of the torsor method. Almost at the same time, R. de la Bretèche [dB98] proved the stronger formula

− 3/5 −1/5 ◦ 7/8 c log B log log B NX ,Hnaive (B)=BQ(log B)+O(B e ) , for Q a polynomial of degree six and c>0. More generally, he developed a method to count the points on possibly singular toric varieties by using the universal torsor [dB01]. Later, R. de la Bretèche reﬁned the asymptotic formula, in joint work with Sir Peter Swinnerton-Dyer [dB/S-D], to

◦ 9/11 13/16+ε NX ,Hnaive (B)=BQ(log B)+γB + O(B ) ,

80 conjectures on points of bounded height [Chap. II albeit only under the assumption of the Riemann hypothesis. Interestingly, the work of de la Bretèche and Swinnerton-Dyer exploits the height zeta function.

8.33. Remarks. i) Recently, U. Derenthal, F. Janda [Der/J]andC.Frei [Fr] proved Manin’s conjecture for the base extension of X to an arbitrary number ﬁeld. Once again, the universal torsor method has been used. Unlike the

situation over É, no higher order terms were established. ii) G. Bhowmik, D. Essouabri, and B. Lichtin [B/E/L] showed the conjecture of Manin for the variety in Pn , given by the equation xn = x · ... · x . É 0 1 n

Part B

The Brauer group

CHAPTER III

On the Brauer group of a scheme

No attention should be paid to the fact that algebra and geometry are diﬀerent in appearance. Omar Khayyám (1070, translated by A. R. Amir-Moez)

This chapter is the technical heart of this book. It is devoted to the concept of an Azumaya algebra over an arbitrary base scheme and, correspondingly, to the Brauer group of an arbitrary base scheme. This is one of the greatest achievements of A. Grothendieck’s refoundation of algebraic geometry [GrBrI, GrBrII, GrBrIII]. Cohomology is an important tool in understanding Brauer groups. As we will recall in Section 1, the Brauer group of a ﬁeld may be written as the Galois cohomology group H2 Gal(Ksep/K), (Ksep)∗ . Analogously, the Brauer group of a scheme is related to an étale cohomology group. In order to follow our reasoning, the reader who is not very familiar with étale cohomology should try to accept the following four main principles. i) For n prime to the characteristic, the Kummer sequence

−→ −→ 0 −→ μn −→ m m 0 is exact with respect to the étale topology. It therefore induces a long exact sequence in cohomology [Mi, Example II.2.18.b)].

i → i ii) For F a torsion sheaf, the are natural isomorphisms Hét(XK ,F) Hét(XK ,F), for K/K an extension of separably closed ﬁelds [Mi, Corollary VI.2.6)]. In par-

ticular, in characteristic zero, one is essentially reduced to the base ﬁeld of the complex numbers.

i ∼ i

iii) For X proper and smooth over , Hét(X, /n ) = Hsing(X, /n ) agrees with the usual singular cohomology [Mi, Theorem III.3.12]. Further, non-canonically,

∼ μn = /n . i iv) If X is deﬁned over an algebraically non-closed ﬁeld K,thenHét(XK ,F) is acted upon by the absolute Galois group Gal(Ksep/K). The arithmetic cohomology group i i Hét(X, F) is obtained by mixing up Hét(XK ,F) with Galois cohomology according to the Hochschild–Serre spectral sequence [Mi, Theorem III.2.20].

Observe that this distinguishes étale from singular cohomology. In fact, on singular 

cohomology, there is such an operation only as long as Gal( /K) acts on X( ) by continuous maps. This essentially enforces that K is Ê (or ). At a few points, we will give references to Grothendieck’s [SGA4], which we consider to be an ideal reference work. It often contains exactly the particular result that we need. Nevertheless, in order to follow the arguments below, an understanding of (parts of) Chapters II and III of Milne’s textbook [Mi] should be suﬃcient.

84 on the brauer group of a scheme [Chap. III

1. Central simple algebras and the Brauer group of a ﬁeld

In this section, we are going to recall the case of a base ﬁeld, which is by far more elementary than the general one. Over ﬁelds, Azumaya algebras are classically called central simple algebras.

1.1. Definition. Let K be any field. Then a central simple algebra over K is a finite dimensional K-vector space A, equipped with a structure of an associative ring with unit such that the following conditions are fulfilled. i) The map K → A, x → x·1, defined via multiplication by scalar, is a homomorphism of rings. ∼ ii) The algebra A is central, i.e., its center coincides with K·1 = K. iii) The algebra A is simple, i.e., it has no two-sided ideals except for (0) and (1).

1.2. Lemma ( J. H. Maclagan-Wedderburn, R. Brauer). Let K be a field. a) Let A be a central simple algebra over K. Then there exist a skew field D with ∼ center K and a natural number n such that A = Mn(D) is isomorphic to the full algebra of n × n-matrices with coefficients in D. b) Let L be a field extension of K and A be a central simple algebra over K.Then A ⊗K L is a central simple algebra over L. c) Assume K to be separably closed. Let D be a skew field being finite dimensional over K whose center is equal to K.ThenD = K.

Proof. See, for example, [Lan93], [Bou-A], or [Ke].

1.3. Remarks. a) Let A be a central simple algebra over a field K. ∼ i) The proof of Lemma 1.2.a) shows that in the presentation A = Mn(D) the skew field D is unique up to isomorphism of K-algebras and the natural number n is unique. sep sep ii) A⊗K K is isomorphic to a full matrix algebra over K . In particular, dimK A is a perfect square. The natural number ind(A):= dimK (D) is called the index of A. b) Let A1,A2 be central simple algebras over a field K.ThenA1 ⊗K A2 can be shown to be a central simple algebra over K.Further,ifA is a central simple op ∼ algebra over a field K,thenA⊗K A = AutK-Vect(A). I.e., it is isomorphic to a matrix algebra. ∼ ∼ c) Two central simple algebras A1 = Mn1 (D1),A2 = Mn2 (D2) over a field K are said to be similar if the corresponding skew fields D1 and D2 are isomorphic as K-algebras. This is an equivalence relation on the set of all isomorphism classes of central simple algebras over K. The tensor product induces a group structure on the set of similarity classes of central simple algebras over K,thisistheso-called Brauer group Br(K) of the field K.

Sec. 1] central simple algebras and the brauer group of a field 85

1.4. Definition. Let K be a field, and let A be a central simple algebra over K. A field extension L of K admitting the property that A⊗K L is isomorphic to a full matrix algebra is said to be a splitting field for A. In this case, one says that A splits over L.

1.5. Lemma (TheoremofSkolemandNoether). Let R be a commutative ring with unit. Then GLn(R) operates on Mn(R) by conjugation,

(g, m) → gmg−1.

If R = L is a ﬁeld, then this deﬁnes an isomorphism ∼ ∗ = PGLn(L):=GLn(L)/L −→ AutL(Mn(L)) .

Proof. One has L = Zent(Mn(L)). Therefore, the mapping is well deﬁned and injective.

Surjectivity. Let j :Mn(L) → Mn(L) be an automorphism. We consider the algebra

op ∼ M := Mn(L) ⊗L Mn(L) (= Mn2 (L)).

Mn(L) gets equipped with the structure of a left M-module in two ways:

(A ⊗ B) •1 C := A · C · B,

(A ⊗ B) •2 C := j(A) · C · B.

2 Two Mn2 (L)-modules of the same L-dimension are isomorphic, as the n -dimensional standard L-vector space equipped with the canonical operation of Mn2 (L) is the only simple left Mn2 (L)-module and there are no non-trivial extensions. Thus, there is an isomorphism h:(Mn(L), •1) → (Mn(L), •2).

Let us put I := h(E) to be the image of the identity matrix. For every M ∈ Mn(L), we have

h(M)=h((E ⊗ M) •1 E)=(E ⊗ M) •2 h(E)=h(E) · M = I · M.

In particular, I ∈ GLn(L). Therefore,

I · M = h(M)=h((M ⊗ E) •1 E)=(M ⊗ E) •2 h(E)=j(M) · I

−1 for each M ∈ Mn(L) and j(M)=IMI .

1.6. Definition. Let n be a natural number. K i) If K is a field, then we will denote by Azn the set of all isomorphism classes of central simple algebras A of dimension n2 over K. L/K ii) Let L/K be a field extension. Then Azn will denote the set of all isomorphism 2 classes of central simple algebras A that are of dimension n over K and split over L. K L/K Obviously, Azn := L/K Azn .

86 on the brauer group of a scheme [Chap. III

1.7. Theorem (cf. J.-P. Serre: Corps locaux [Se62, chap. X, §5]). Let L/K be a ﬁnite Galois extension of ﬁelds, let G := Gal(L/K) be its Galois group, and ∈ let n Æ. Then there is a natural bijection of pointed sets

∼ L/K L/K −→= 1 a = an :Azn H (G, PGLn(L)) ,

A → aA .

Proof. Let A be a central simple algebra over K that splits over L, ∼ = A ⊗K L −→ Mn(L) . f The diagrams ⊗ f / A OK L MnO(L)

σ σ f / A ⊗K L Mn(L) do not commute, in general.

For each σ ∈ G, deﬁne aσ ∈ PGLn(L) by putting (f ◦ σ)=aσ ◦ (σ ◦ f). It turns out that

f ◦ στ =(f ◦ σ) ◦ τ

= aσ ◦ (σ ◦ f) ◦ τ

= aσ ◦ σ ◦ (f ◦ τ)

= aσ ◦ σ ◦ (aτ ◦ (τ ◦ f)) σ = aσ ◦ aτ ◦ (στ ◦ f) .

σ I.e., aστ = aσ · aτ and (aσ)σ∈G is a cocycle. If one starts with another isomorphism f : A ⊗K L −→ Mn(L), then there exists some b ∈ PGLn(L) such that f = b ◦ f . The equality (f ◦ σ)=aσ ◦ (σ ◦ f) implies

−1 −1 −1 σ f ◦ σ = b ◦ f ◦ σ = b · aσ ◦ (σ ◦ (b ◦ f )) = b · aσ · b ◦ (σ ◦ f ) .

Thus, the isomorphism f yields a cocycle cohomologous to (aσ)σ∈G. The mapping a is well deﬁned. Injectivity. Assume that A and A are chosen in such a way that the construction 1 above yields one and the same cohomology class aA = aA ∈ H (G, PGLn(L)). After choosing suitable isomorphisms f and f , one has the equalities (f ◦ σ)=aσ ◦ (σ ◦ f) and (f ◦ σ)=aσ ◦ (σ ◦ f ) in the diagram

⊗ f / o f ⊗ A OK L MnO(L) A OK L

σ σ σ f / o f A ⊗K L Mn(L) A ⊗K L.

Sec. 1] central simple algebras and the brauer group of a field 87

Consequently, f ◦ σ ◦ f −1 ◦ σ−1 = f ◦ σ ◦ f −1 ◦ σ−1 and, therefore,

f ◦ σ ◦ f −1 ◦ f ◦ σ−1 ◦ f −1 =id.

The outer part of the diagram commutes. Taking the G-invariants on both sides ∼ yields A = A . 1 Surjectivity. Let a cocycle (aσ)σ∈G for H (G, PGLn(L)) be given. We deﬁne a new G-operation on Mn(L) as follows. Let σ ∈ G act as σ aσ aσ ◦ σ :Mn(L) −→ Mn(L) −→ Mn(L) .

Note that this is a σ-linear mapping. Further, one has

σ (aσ ◦ σ) ◦ (aτ ◦ τ)=aσ ◦ aτ ◦ στ = aστ ◦ στ .

I.e., we constructed a group operation from the left. Galois descent yields the desired algebra.

1.8. Corollary. Let L/K be a finite Galois extension of fields, and let n be a natural number. a) Let L be a field extension of L such that L/K is Galois, too. Then the following diagram of morphisms of pointed sets commutes,

aL/K L/K n / 1 Azn H (Gal (L/K), PGLn(L))

Gal (L /K) nat. incl. infGal(L/K)

 aL /K L /K n / 1 Azn H Gal(L /K), PGLn(L ) . b) Let K be an intermediate ﬁeld of the extension L/K. Then the following diagram of morphisms of pointed sets commutes,

aL/K L/K n / 1 Azn H Gal(L/K), PGLn(L)

⊗ Gal(L/K ) KK resGal(L/K)

 aL/K L/K n / 1 Azn H Gal(L/K ), PGLn(L) . ∗ Proof. These are direct consequences of the construction of the bijections an.

1.9. Corollary. Let K be a ﬁeld, and let n be a natural number. Then there is a unique natural bijection K K −→ 1 sep sep a = an :Azn H Gal(K /K), PGLn(K )

88 on the brauer group of a scheme [Chap. III

K L/K sep such that a | L/K = an for each finite Galois extension L/K in K . n Azn Proof. In order to get connected to the definition of the cohomology of a profinite group, only one technical point is to be proven. That is the formula

sep Gal(Ksep/K) PGLn(K ) =PGLn(K ) for K ⊆ K ⊆ Ksep any intermediate ﬁeld. To see this, observe that the exact sequence sep ∗ sep sep 1 −→ (K ) −→ GLn(K ) −→ PGLn(K ) −→ 1 induces a long exact sequence ∗ sep Gal(Ksep/K) 1 sep sep ∗ 1→(K ) → GLn(K ) → PGLn(K ) → H Gal(K /K ), (K ) in cohomology. Finally, the right entry vanishes by Hilbert’s Theorem 90 [Gru, Proposition 3].

1.10. Proposition. Let K be a ﬁeld, and let m and n be natural numbers. Then there is a commutative diagram

aK K n / 1 sep sep Azn H Gal(K /K), PGLn(K )

→ n A Mm(A) (imn)∗

 aK K mn / 1 sep sep Azmn H Gal(K /K), PGLmn(K ) .

n Here, (imn)∗ is the map induced by the block-diagonal embedding

n sep sep i :PGLn(K ) −→ PGLmn(K ) mn ⎛ ⎞ E 0 ··· 0 ⎜ ⎟ ⎜ 0 E ··· 0 ⎟ E → ⎜ . . . . ⎟ . ⎝ . . .. . ⎠ 00··· E

∈ K Proof. Let A Azn . By the construction above, a cycle represent- K ing the cohomology class an (A) is given as follows. Choose an isomorphism sep sep −1 sep f : A ⊗K K → Mn(K ), and put aσ := (f ◦ σ) ◦ (σ ◦ f) ∈ Aut(Mn(K )) for each σ ∈ Gal(Ksep/K). ∈ K On the other hand, for Mm(A) Azmn one may choose the isomorphism

sep sep sep ∼ sep Mm(f): Mm(A) ⊗K K =Mm(A ⊗K K ) −→ Mm(Mn(K )) = Mmn(K ).

sep sep For each σ ∈ Gal(K /K), this yields the automorphism 0aσ of Mm(Mn(K )), which operates as aσ on each block. If aσ is given by conjugation with a matrix

Sec. 2] azumaya algebras 89

A ,then0a is given by conjugation with σ σ ⎛ ⎞ Aσ 0 ··· 0 ⎜ ⎟ ⎜ 0 Aσ ··· 0 ⎟ ⎜ . . . . ⎟ . ⎝ . . .. . ⎠ 00··· Aσ This is exactly what was to be proven.

1.11. Remark. The proposition above shows ∼ 1 sep sep Br(K) = −→lim H Gal(K /K), PGLn(K ) . n Further, for each m and n, there is a commutative diagram of exact sequences as follows, / sep ∗ / sep / sep / 1 (K ) GLn(K ) PGLn(K ) 1

jn mn / sep ∗ / sep / sep / 1 (K ) GLmn(K ) PGLmn(K ) 1 . sep ∗ sep sep We note that (K ) is mapped into the centers of GLn(K ) and GLmn(K ),respectively. Therefore, there are boundary maps to the second group cohomology group and they are compatible with each other to give a map 1 sep sep −→ 2 sep sep ∗ −→lim H Gal(K /K), PGLn(K ) H Gal(K /K), (K ) . n It is not complicated to show that this map is injective and surjective. Cf. Corollary 6.3, below.

1.12. Remark. It is clearly of importance to know the Brauer group explicitly for particular fields. Such computations are typically not easy. For example, for number fields, they emerge as a byproduct of global class field theory. We will come back to this in Section 8.

2. Azumaya algebras

2.1. Definition. Let X be any scheme. Then a sheaf of Azumaya algebras or simply an Azumaya algebra over X is a locally free sheaf A of locally finite rank over X equipped with the structure of a sheaf of OX -algebras such that the following condition is fulfilled.

For every closed point x ∈ X, one has that A (x):=A ⊗X k(x) is a central simple algebra over the residue ﬁeld k(x).

2.2. Example. If X = Spec k is the spectrum of a ﬁeld, then an Azumaya algebra over X is nothing but a central simple algebra over k.

90 on the brauer group of a scheme [Chap. III

2.3. Proposition. Let X be any scheme, and let A be any locally free sheaf of locally ﬁnite rank on X having the structure of a sheaf of algebras. Then A is an Azumaya algebra over X if and only if the canonical homomorphism

op ιA : A ⊗X A −→ End(A ) , a ⊗ b → (x → axb) is an isomorphism of locally free sheaves. Here, a, b, x ∈ A (U) denote sections of A over an arbitrary open subset U of X.

Proof. “=⇒”LetA be an Azumaya algebra. Then A (x) is a central simple algebra over k(x) for every closed point x ∈ X.ThisimpliesthatιA is an isomorphism at every closed point. It is therefore an isomorphism of locally free sheaves. “⇐=”Letx ∈ X be a closed point. We need to show that A (x) is a central simple algebra over k(x). By assumption, we know that

op ∼ n ∼ A (x) ⊗k(x) [A (x)] = End(k(x) ) = Mn(k(x)) for n := rkx(A ).Further,Mn(k(x)) is a central simple algebra over k(x). If A (x) were not central, Z(A (x)) = L k(x),then

op ∼ L = L ⊗k(x) k(x) ⊆ A (x) ⊗k(x) [A (x)] = Mn(k(x)) would be contained in the center of Mn(k(x)). This is a contradiction. Assume, ﬁnally, A (x) is not simple. Then it would contain a proper two-sided op ideal (0) I A (x). But, under this assumption, I ⊗k(x) [A (x)] would be a op ∼ non-trivial, proper two-sided ideal in A (x) ⊗k(x) [A (x)] = Mn(k(x)). We obtained a contradiction. A is, therefore, a central simple algebra.

2.4. Deﬁnition. Let X be any scheme, and let A be a locally free sheaf on X having the structure of a sheaf of algebras. a) Then, for a point x ∈ X, we will say that A is Azumaya in x if A (x) is a central simple algebra over k(x). b) The set of all points x ∈ X such that A is not Azumaya in x will be called the non-Azumaya locus of A .

2.5. Corollary. Let X be any scheme, and let A be any locally free sheaf on X having the structure of a sheaf of algebras. a) Then the non-Azumaya locus T ⊂ X of A is the support of a Cartier divisor. In particular, it is a closed set in X. b) Assume, X is regular and connected. Then the non-Azumaya locus of A is either the whole of X or a subset pure of codimension one.

Sec. 2] azumaya algebras 91

Proof. a) The assertion is local in X. We may, therefore, assume that A is a free sheaf, say, of rank n.

A ⊗ A op ∼ On2 −→ E A ∼ On2 ιA : X = X nd( ) = X is then a morphism of free sheaves, which are both of rank n2.

Whether ιA ⊗X k(x) is an isomorphism is equivalent to the non-vanishing of its determinant in x. In the trivialization, the latter is a section s of the structure sheaf OX .divs is, by deﬁnition, the support of a Cartier divisor. b) This is an immediate consequence of a).

2.6. Fact. Let A and B be two Azumaya algebras on a scheme X. A ⊗ B Then their tensor product OX as locally free sheaves over X, equipped with the obvious structure of an OX -algebra, is again an Azumaya algebra. Proof. On sections over an open subset U ⊆ X, the algebra structure is given by

(a ⊗ b)(a ⊗ b):=aa ⊗ bb.

A ⊗ B O It is clear that OX is a locally free X -module. The property of being Azumaya may be tested in closed points. One has

A ⊗ B A ⊗ B⊗ A ⊗ ⊗ B⊗ ( OX )(x)= OX OX k(x)=( OX k(x)) k(x) ( OX k(x)) .

On the right-hand side, the tensor product of two central simple algebras is again a central simple algebra.

2.7. Fact. Let f : X → Y be a morphism of schemes, and let A be an Azumaya algebra on Y . Then the pullback f ∗A of A as a locally free sheaf, equipped with the obvious structure of an OX -algebra, is an Azumaya algebra on X. Proof. As the assertion is local in both Y and X, we may assume that f : X = Spec S → Y = Spec R is a morphism of aﬃne schemes and that A is given by an R-algebra A that is free of ﬁnite rank as an R-module. ∗ Then f A is given by the S-algebra A⊗R S. Its algebra structure is given, analogously to the one on the tensor product above, by

(a ⊗ s)(a ⊗ s):=aa ⊗ ss.

Again, we test the property of being Azumaya in closed points. For that, let x ∈ X, and put y := f(x).Thus,k(x) is a ﬁeld extension of k(y).

What we have to show is that A⊗RS ⊗S k(x) is a central simple algebra over k(x). But, A⊗R S ⊗S k(x)=A⊗R k(x)=A⊗R k(y)⊗k(y) k(x) and A⊗R k(y) is, by assumption, a central simple algebra over k(y). The assertion follows.

92 on the brauer group of a scheme [Chap. III

2.8. Proposition. Let f : X → Y be a morphism of schemes that is faithfully flat. Further, let A be a quasi-coherent sheaf on Y , equipped with the structure of a sheaf of OY -algebras. Then A is an Azumaya algebra on Y if and only if f ∗A is an Azumaya algebra on X. Proof. “=⇒”ThisisFact2.7. “⇐=” The assumption that f ∗A is an Azumaya algebra on X includes that f ∗A is a locally free OX -module, locally of finite rank. We aim first at showing that this implies A is a locally free OY -module, locally of finite rank. For that, let y ∈ Y ,andchoosex ∈ X such that f(x)=y.Wehaveaflatlocal homomorphism of local rings i: OY,y → OX,x.By[Mat, Theorem 7.2], OX,x is faithfully flat over OY,y. ∗A A ⊗ O O Further, (f )x = y OY,y X,x is a locally free X,x-module of finite rank. Hence, Ay is indeed a locally free OY,y-module and of finite rank. This means, A is a locally free OY -module, locally of finite rank. In the remainder of the proof, flatness will no longer be used. To verify that A is an Azumaya algebra over Y , we will use the criterion provided ∗A A ⊗ O by Proposition 2.3. We have f = OY X and, therefore, ∗A ⊗ ∗A op A ⊗ O ⊗ O ⊗ A op f OX [f ] =( OY X ) OX ( X OY [ ] ) ∼ A ⊗ A op ⊗ O = OY [ ] OY X ∗ A ⊗ A op = f ( OY [ ] ) .

Further, End(f ∗A )=f ∗(End(A )), and the isomorphisms are compatible in such a way that

∗ f ιA = ιf ∗A .

∗ Thus, we are given that f ιA = ιf ∗A is an isomorphism of locally free sheaves. We have to prove this implies that the original ιA was an isomorphism. ∗ # # For that, note that det(f ιA )=f det(ιA ).Thus,weknowthatf det(ιA ) is a nowhere vanishing section of the invertible sheaf

∗ max A ⊗ A op ∨ ⊗ max E A f [Λ ( OY [ ] ) Λ ( nd( ))], and we have to show that det(ιA ) is nowhere vanishing. Again, for y ∈ Y , we choose x ∈ X such that f(x)=y. Thenwehavealocal homomorphism of local rings i: OY,y → OX,x. As we work in stalks, we may choose a trivialization and write det(ιA ) ∈ OY,y.Then

# f det(ιA )=i(det(ιA )) ∈ OX,x .

If det(ιA ) were vanishing at y, then this would simply mean det(ιA ) ∈ mY,y.Then # i(det(ιA )) ∈ mX,x. I.e., f det(ιA ) vanishes at x. This is a contradiction.

Sec. 3] the brauer group 93

2.9. Proposition. Let f : X → Y be a morphism of schemes that is faithfully ﬂat and quasi-compact. Then it is equivalent to give a) an Azumaya algebra A over Y , b) an Azumaya algebra B over X, together with an isomorphism

∗ B → ∗ B Φ: pr1 pr2 of Azumaya algebras on X ×Y X satisfying

∗ ∗ ◦ ∗ pr31(Φ) = pr32(Φ) pr21(Φ) on X ×Y X ×Y X. Sketch of Proof. “a) =⇒ b)” Put B := f ∗A . “b) =⇒ a)” This is what is called faithful ﬂat descent. ∗ The existence of A as a quasi-coherent OY -module satisfying B := f A follows directly from [K/O74, Theorem II.3.2]. The additional algebra structure descends, too. This may easily be seen from the construction of the descent module givenintheproof.Itisshownin[K/O74, Theorem II.3.4]. A is an Azumaya algebra over Y by virtue of Proposition 2.8.

2.10. Remark. This means that an Azumaya algebra over a scheme X may be described by local gluing data with respect to the fpqc-topology or any weaker topology. Intuitively, B describes the desired Azumaya algebra on each set of a cover. Φ collects the gluing isomorphisms on intersections of two sets of the cover. The condition on X ×Y X ×Y X is a cocycle condition encoding that the gluing maps be compatible. We will use this approach for constructing Azumaya algebras from data local in the étale topology.

3. The Brauer group

3.1. Deﬁnition. Let X be any scheme. Two Azumaya algebras A and B on X are said to be similar if there exist two locally free OX -modules E and F , both everywhere of positive rank, such that

A ⊗ E E ∼ B ⊗ E E OX nd( ) = OX nd( ) .

3.2. Remarks. i) Similarity is an equivalence relation. For that, the only point that is not entirely obvious is transitivity. To achieve this, assume

A ⊗ E E ∼ B ⊗ E E B ⊗ E E ∼ C ⊗ E E OX nd( ) = OX nd( )and OX nd( ) = OX nd( ) .

94 on the brauer group of a scheme [Chap. III

Then

A ⊗ E E ⊗ E ∼ A ⊗ E E ⊗ E E OX nd( OX ) = OX nd( ) OX nd( ) ∼ B ⊗ E E ⊗ E E = OX nd( ) OX nd( ) ∼ B ⊗ E E ⊗ E E = OX nd( ) OX nd( ) ∼ C ⊗ E E ⊗ E E = OX nd( ) OX nd( ) ∼ C ⊗ E E ⊗ E = OX nd( OX ) . ii) Similarity is compatible with the tensor product of Azumaya algebras and with pullback.

3.3. Deﬁnition. Let X be any scheme. The set of all similarity classes of Azumaya algebras on X is called the Brauer group of X and denoted by Br(X).

3.4. Remarks. i) The similarity class of an Azumaya algebra A will be denoted by [A ]. A ∗ A A ⊗ A ii) A binary operation on Br(X) is given by [ ] [ ]:=[ OX ] . iii) For this binary operation, [End(E )] is the neutral element. Here, E is an arbitrary locally free sheaf, nowhere of rank zero. iv) The inverse element of [A ] is given by [A op]. This shows that Br(X) is indeed a group.

4. The cohomological Brauer group

4.1. Lemma. Let (R, m) be a Henselian local ring, and let A be an Azu- maya algebra over Spec R. Assume

A ⊗ O ∼ E On OSpec R Spec R/m = nd( Spec R/m)

∈ A ∼ n Æ E O for a certain n .Then = nd( Spec R). Proof. We denote by A the R-algebra corresponding to A . In this notation, ∼ we, therefore, have A ⊗R R/m = Mn(R/m). On the right-hand side choose an idempotent matrix of rank one. Let a ∈ A be such that a := (a mod mA)=.ThenR[a] is a ﬁnite commutative R-algebra. As R is Henselian, [SGA4,Exp.VIII,4.1]showsthatR[a] is a direct ∼ ∼ product of local rings. From R[a]⊗Rk = k[] = k ⊕k,weseethatR[a] = R/I ⊕R/J is actually composed of two local R-algebras. In k ⊕ k,let correspond to the element (1, 0). Thus, the element e ∈ R[a] mapped to (1, 0) under the isomorphism R[a] → R/I ⊕ R/J is an idempotent lifting . Consider the homomorphism of R-algebras

φ: A −→ EndR(Ae) , a → (be → abe) .

Sec. 4] the cohomological brauer group 95

Both R-modules are free of the same rank. It is classically known that φ⊗R k is an isomorphism. Nakayama’s lemma ensures that φ is an isomorphism itself.

4.2. Corollary. Let (R, m) be a strictly Henselian local ring, and let A be an Azumaya algebra over Spec R. A ∼ E On Then = nd( Spec R). Proof. k = R/m is a separably closed ﬁeld. We know that

A ⊗ O ∼ E On OSpec R Spec R/m = nd( Spec R/m) is a full matrix algebra over R/m.

4.3. Lemma (Theorem of Skolem and Noether, version over local rings). Let R be a commutative ring with unit. Then GLn(R) operates on Mn(R) by conjugation, (g, m) → gmg−1.

If R =(R, m) is a local ring, then this deﬁnes an isomorphism ∼ ∗ = PGLn(R):=GLn(R)/R −→ AutR(Mn(R)) .

Proof. One has R = Zent(Mn(R)). Therefore, the mapping is well deﬁned and injective.

Surjectivity. Let j :Mn(R) → Mn(R) be an automorphism. We consider the algebra

op ∼ ∼ M := Mn(R) ⊗R Mn(R) (= EndR-mod(Mn(R)) = Mn2 (R)) .

Mn(R) gets equipped with the structure of a left M-module in two ways:

(A ⊗ B) •1 C := A · C · B,

(A ⊗ B) •2 C := j(A) · C · B.

We will denote the resulting M-modules by N1 and N2, respectively.

Our ﬁrst claim is that N1 is a projective M-module. This means, we have to show that Mn(R) is projective when equipped with its canonical structure as an EndR-mod(Mn(R))-module.

For that, consider a morphism i:Mn(R) → R of R-modules that is a section of the canonical inclusion mapping r to r·E. Note that such a section exists because Mn(R) is a projective R-module.

Then the surjection of EndR-mod(Mn(R))-modules

EndR-mod(Mn(R)) → Mn(R), given by f → f(1), admits the section

m → (m → i(m)m) .

96 on the brauer group of a scheme [Chap. III

This shows Mn(R) is a projective EndR-mod(Mn(R))-module.

Over the ﬁeld R/m, it is known that two Mn2 (R/m)-modules of the same R/m- dimension are isomorphic. Thus, there is an isomorphism

g : M1 ⊗R R/m → M2 ⊗R R/m .

As M1 is projective, the map

π g M1 −→ M1 ⊗R R/m −→ M2 ⊗R R/m may be lifted to a homomorphism h: M1 → M2 of EndR-mod(Mn(R))-modules. h⊗RR/m is an isomorphism. Nakayama’s lemma implies that h is an isomorphism itself.

Let us put I := h(E) to be the image of the identity matrix. For every M ∈ Mn(R), we have

h(M)=h((E ⊗ M) •1 E)=(E ⊗ M) •2 h(E)=h(E) · M = I · M.

In particular, I ∈ GLn(R). Therefore,

I · M = h(M)=h((M ⊗ E) •1 E)=(M ⊗ E) •2 h(E)=j(M) · I

−1 for each M ∈ Mn(R) and j(M)=IMI .

4.4. Proposition Let X be any quasi-compact scheme. Then the set of all isomorphism classes of rank n2 Azumaya algebras on X is given by the Čech coho- ˇ 1 mology set Hét(X, PGLn). Proof. We know by Proposition 2.9 that Azumaya algebras may be described by gluing data local in the étale topology. Let A be any Azumaya algebra over X. For every point x ∈ X, Corol- lary 4.2 yields that A is isomorphic to matrix algebra after strict Henselization, A ⊗ Osh ∼ Osh OX X,x = Mn( X,x). To describe the isomorphism, only finitely many data Osh on X,x are necessary. For this reason, there exists an affine scheme Ux,which is an étale neighbourhood of x, such that A | ∼ End(On ).Byvirtueofquasi- Ux = Ux compactness, one may select finitely many of the Ui covering the whole of X. A All these1 étale1 neighbourhoods together form gluing data for since the morphism → Ux1 ... Uxl X is faithfully flat and quasi-compact. The conditions for gluing data formulated in Proposition 2.9 are equivalent to giving ˇ 1 aČechcocycleforHét(X, Aut(Mn)). Different gluing data describing the same algebra differ by a coboundary.

The Theorem of Skolem and Noether implies that the sheaf Aut(Mn) is the same as PGLn.

4.5. Remark. On a quasi-compact scheme X, the set of all isomorphism classes of rank n locally free sheaves is given by the Čech cohomology set ˇ 1 Hét(X, GLn). This is proven in the same way as Proposition 4.4.

Sec. 4] the cohomological brauer group 97

4.6. Notation. Let X be any scheme. i) We will denote the cohomology class corresponding to an Azumaya algebra A A ∈ ˇ 1 over X by c( ) Hét(X, PGLn). ii) We will denote the cohomology class corresponding to a locally free sheaf E E ∈ ˇ 1 over X by cl( ) Hét(X, GLn). iii) Recall that there is a short exact sequence

−→ −→ −→ −→ 0 m GLn PGLn 0 of sheaves. In cohomology, it induces a long exact sequence of pointed sets

ˇ 1 ι ˇ 1 d 2 −→ −→ Hét(X, GLn) Hét(X, PGLn) Hét(X, m).

4.7. Facts. Let X be a quasi-compact scheme. i) For a locally free sheaf E on X, one has that

ι(cl(E )) = c(End(E )) . ii) For two Azumaya algebras A and B on X of ranks n and m, respectively, one has A ⊗ B A ∗ B c( OX )=c( ) c( ) .

Here, the operation ∗ is induced by the canonical mapping

PGLn × PGLm −→ PGLnm given by

n m n m nm

× × −→ ⊗ GLn GLm = Aut( a ) Aut( a ) Aut( a a )=Aut( a )=GLnm .

∈ ˇ 1 ∈ ˇ 1 iii) For α Hét(X, PGLn) and β Hét(X, PGLn), we have

d(α∗β)=d(α)+d(β) .

Proof. The veriﬁcation of these compatibilities is purely technical and hence omitted.

4.8. Proposition. Let X be any quasi-compact scheme. Then the boundary maps d induce an injective group homomorphism

2 −→ iX : Br(X) Hét(X, m) .

98 on the brauer group of a scheme [Chap. III

Proof. First step. iX is well deﬁned. Suppose we are given two Azumaya algebras A and B over X that are similar. This means we have two locally free sheaves, E and F , such that A ⊗ E E ∼ B ⊗ E E OX nd( ) = OX nd( ).Inparticular, A ⊗ E E B ⊗ E E ∗ c( OX nd( )) = c( OX nd( )) . ( )

The compatibilities above yield

A ⊗ E E A ∗ E E A ∗ E c( OX nd( )) = c( ) c( nd( )) = c( ) ι(cl( )) , hence

A A E A ∗ E A ⊗ E E d(c( )) = d(c( )) + d(ι(cl( ))) = d(c( ) ι(cl( ))) = d(c( OX nd( ))) . B B ⊗ E E Completely analogously, one shows that d(c( )) = d(c( OX nd( ))) . For- mula (∗), therefore, yields the claim.

Second step. iX is a group homomorphism. A ⊗ B A ∗ B A B We have d(c( OX )) = d(c( ) c( )) = d(c( )) + d(c( )) .

Third step. iX is an injection. Suppose, we have an Azumaya algebra A such that d(c(A )) = 0. Then, according to the exactness of the long cohomology sequence, c(A )=ι(cl(E )) for a locally ∼ free sheaf E on X.Thus,A = End(E ) and [A ]=0∈ Br(X).

2 4.9. Deﬁnition. Let X be any scheme. Then Hét(X, m) is called the cohomological Brauer group of X. It will be denoted by Br(X).

4.10. Remark. iX is, in general, not an isomorphism of groups. The ﬁrst counterexample to surjectivity has been constructed by A. Grothendieck in [GrBrII, Remarque 1.11.b]. iX is, however, an isomorphism in a number of particular cases. One such case is that of smooth algebraic surfaces. This was essentially known to M. Auslander and O. Goldman [A/G] before Grothendieck’s invention of the general Brauer group for schemes, perhaps even before the actual invention of the concept of a scheme. In Section 7, we will present the result of Auslander and Goldman in a more up- to-date formulation. Although this is not the most general result known today, we hope it may serve as a good illustration for what Br(X) and Br (X) are and which methods might be used in order to compare them.

5. The relation to the Brauer group of the function ﬁeld

5.1. In this section, we will always assume that X is an integral scheme. Denote by g : Spec K = η → X the inclusion of the generic point.

Sec. 5] the relation to the brauer group of the function field 99

5.2. Fact. Let X be an integral scheme that is regular, separated, and quasi- compact. Then Br(X) ⊆ Br(X) ⊆ Br(K) , where K := Q(X) denotes the function ﬁeld of X. Proof. Denote by g : η → X the inclusion of the generic point. We have the short exact sequence

−→ −→ −→ −→ 0 m,X g∗ m,K DivX 0 of sheaves in the étale topology. Here, DivX = codim x=1 ix∗ x, which implies

1 1

{ } Hét(X, DivX )= Hét( x , )= Hom(Gx, )=0. codim x=1 codim x=1 It follows, at ﬁrst, that

2 2 ⊆ Hét(X, m) Hét(X, g∗ m,K ).

Second, we consider the Leray spectral sequence

p,q p q n ⇒ E2 := Hét(X, R g∗ m,K )= Hét(Spec K, m,K ).

By virtue of Hilbert’s Theorem 90, for the ﬁrst higher direct image, we have 1 R g∗ m,K =0. This implies there is an exact sequence

2 2 0 2

−→ −→ † −→ 0 Hét(X, g∗ m,K ) Hét(Spec K, m,K ) Hét(X, R g∗ m,K ) ( ) of terms of lower order.

5.3. Remark. One may therefore think of Br(X) as follows. It consists of those classes of the usual Br(Q(X)) that allow an extension over the whole of X.

5.4. Lemma. Let X be an integral scheme that is separated and quasi- q compact. Then, for each q>0,thesheaf R g∗ m,K is uniquely l-divisible for ∈ every l prime to the residue characteristics of X. Proof. The inclusion g : η → X is the inverse limit, in the category of all X- schemes, of the open embeddings gU : U → X for U ⊆ X aﬃne open. The assump- ⊆ tion that X is separated makes sure that all the transition maps gU ,U : U −→ U are aﬃne. In this situation, from [SGA4, Exp. VII, Corollaire 5.11], we see that

q q R g∗μl =− lim→ R gU∗μl . U

As the gU are open embeddings, and hence smooth, they are acyclic [SGA4, Exp. XV, Théorème 2.1]. In particular, for q>0 and l prime to the residue q q ∗ characteristics of X,wehaveR gU∗μl = R gU∗gU μl,X =0. Altogether, q R g∗μl =0.

100 on the brauer group of a scheme [Chap. III

The Kummer sequence implies that the multiplication map

q q → l : R g∗ m,K R g∗ m,K is an isomorphism for every q>0.

5.5. Proposition. Let X be an integral scheme that is regular, separated, and quasi-compact. 0 2 a) If char(Q(X)) = p>0,thenHét(X, R g∗ m,K )tors is a p-power torsion group. 0 2 b) If all residue characteristics of X are equal to zero, then Hét(X, R g∗ m,K ) is torsion-free. 2 Proof. b) For every l>0,thesheafR g∗ m,K is uniquely l-divisible. There-

0 2 0 2 → fore, the multiplication map l : Hét(X, R g∗ m,K ) Hét(X, R g∗ m,K ) is an isomorphism. For

0 2 0 2 → l : Hét(X, R g∗ m,K )tors Hét(X, R g∗ m,K )tors

0 2 still the same is true. This clearly implies that Hét(X, R g∗ m,K )=0. a) Here, the same argument still works for l prime to p. This shows that 0 2 Hét(X, R g∗ m,K ) has no prime-to-p torsion.

5.6. Corollary. Let X be an integral scheme that is regular, separated, and quasi-compact. Assume that the residue characteristics of X are equal to zero. Then, for g : η → X the inclusion of the generic point, one has

2 ∼ 2 Hét(X, g∗ m,K ) = Hét(Spec K, m,K ) .

2 2 ∗ Proof. Observe that Hét(Spec K, m,K )=H (Gal(K/K), K ) is a torsion group directly by its deﬁnition. Hence, the assertion follows directly from the exact sequence (†) together with Proposition 5.5.b).

5.7. Remark. The same result is true for schemes of dimension at most one in any characteristic as is often shown in the literature. See, e.g., [GrBrIII, formule (2.2)].

5.8. Proposition. Let X be an integral scheme that is regular, separated, and quasi-compact. Assume that i) dim X ≤ 1 or ii) the residue characteristics of X are equal to zero. Then there is a short exact sequence

−→ 0 Br (X) −→ Br (Spec Q(X)) −→ Hom(Gx, É/ ) . codim x=1 Proof. Consider once more the short exact sequence

−→ −→ −→ −→ 0 m,X g∗ m,K DivX 0

Sec. 6] the brauer group and the cohomological brauer group 101

1 of sheaves in the étale topology. Since Hét(X, DivX )=0, as shown above, we have the following fragment of the long exact sequence

2 2 2

−→ −→ −→ 0 Hét(X, m,X ) Hét(X, g∗ m,K ) Hét(X, DivX )

in cohomology. Here, the entry on the left-hand side is Br (X),accordingtoits very deﬁnition. The term in the middle is isomorphic to Br(Q(X)) as was shown in Corollary 5.6. For the term on the right-hand side, we have

2 2 { } Hét(X, DivX )= Hét( x , ) codimx=1 2 = H (Gx, )

codimx=1

= Hom(Gx, É/ ) . codim x=1

5.9. Corollary. Let X be an integral scheme that is regular, separated, and quasi-compact. Assume that i) dim X ≤ 1 or ii) the residue characteristics of X are equal to zero.

Then one has 2 Br (X)= Br (Spec OX,x), codim x=1 where the intersection takes place in Br (Q(X)).

Proof. “⊆” is clear. “⊇” From Proposition 5.8, we see that all obstructions against an extension of a central simple algebra over Q(X) to the whole of X are given by the points of codimension one. For Spec OX,x instead of X, the same is true.

5.10. Remark. Under certain hypotheses on X, there is the same formula for the Brauer group instead of the cohomological Brauer group. Compare Theo- rem 7.9.i), below.

6. The Brauer group and the cohomological Brauer group

6.1. In order to compare the two Brauer groups in relatively simple cases, the following lemma is helpful.

6.2. Lemma. Let X = Spec R for a local ring R,andletγ ∈ Br(X) be an element in the cohomological Brauer group. Assume that there exists a morphism π : Y → X of schemes that is ﬁnite and faithfully ﬂat such that γ ∈ ker(Br(X) → Br(Y )). Then one has even γ ∈ Br(X).

102 on the brauer group of a scheme [Chap. III ∼ Proof. We have Y = Spec S. Nakayama’s lemma implies that S = Rn is free as an R-module. As π is ﬁnite, it has no higher direct images and the Leray spectral sequence collapses to

2 2 Hét(Y, m)=Hét(X, π∗ m) .

We also have the long exact sequence

1 2 2

 −→ −→ Hét(X, π∗ m/ m) Hét(X, m) Hét(X, π∗ m)

2 at our disposal. γ vanishes in Hét(X, π∗ m). It belongs, therefore, to the image

1 of Hét(X, π∗ m/ m). ∗ n The canonical map S → Aut(R )=GLn(R) induces a homomorphism of sheaves

→ π∗ m GLn .

Finally, we note that the diagram

1 / 2

 Hét(X, π∗ m/ m) Hét(X, m)

ˇ 1 /ˇ 2 Hét(X, GLn/m) Hét(X, m) ˇ 1 ˇ 1 commutes. Hence, γ is in the image of Hét(X, GLn/m)=Hét(X, PGLn).

6.3. Corollary. One has Br(Spec K)=Br (Spec K) for every ﬁeld K. 2 ∗ Proof. Let γ ∈ Br (Spec K)=H Gal (K/K), K . Then there is a ﬁnite Ga- lois extension L/K such that γ ∈ H2 Gal(L/K),L∗ , already. When restricted to Spec L, the cohomology class γ vanishes.

6.4. Corollary. One has Br(Spec R)=Br(Spec R)=Br(Spec R/m) for every Henselian local ring (R, m).Inparticular,Br(Spec R)=0for R strictly local. Proof. The equality Br(Spec R)=Br(Spec R/m) follows directly from a general result on étale cohomology [SGA4, Exp. VIII, Corollaire 8.6]. Thus, let γ ∈ Br(Spec R). For its image in Br(Spec R/m), choose a splitting field l that is finite over k := R/m.Leta be a primitive element, and let f ∈ k[X] ∼ be its minimal polynomial. Then l = k[X]/(f). Choose a lift F ∈ R[X] of f. The local ring (S, n):=R[X]/(F ) is finite and flat over R. S is Henselian by [SGA4, Exp. VIII, 4.1.ii)]. Under pullback to Spec S, the cohomology class γ vanishes since Br(Spec S)=Br(Spec S/n).

6.5. Corollary. One has Br(Spec R)=Br(Spec R) for every discrete valuation ring R. Proof. Let γ ∈ Br (Spec R) ⊆ Br (Spec Q(R)). We know that γ vanishes after a finite, separable field extension L/Q(R). Take for S the integral closure of R in L. Then γ vanishes in Br(Spec S) ⊆ Br(L) and S is finite and flat over R.

Sec. 7] the theorem of auslander and goldman 103

6.6. Remark. For X an integral scheme that is regular, separated, and quasi-compact in characteristic zero, one therefore has 2 Br (X)= Br(Spec OX,x) codim x=1 in Br(Q(X)).

7. The theorem of Auslander and Goldman i. Orders over a general scheme.

7.1. Deﬁnition. Let X be an integral scheme, and let A be a central simple algebra over its quotient ﬁeld Q(X).

By an order in A over X, one means an OX -algebra A that is coherent as an OX - module and has A as its stalk at the generic point.

7.2. Remark. Assume the integral scheme X is locally Noetherian. Then, for every central simple algebra A over Q(X), there exists an order over X.

Indeed, an order may be constructed as follows. Choose a set {a1, ... ,al} of generators of A as a Q(X)-vector space. In the sheaf g∗A on X, these generate a coherent subsheaf G . Then the sheaf (G : G ) ⊆ g∗A, that is deﬁned by

(G : G )(U):={ s ∈ g∗A(U) | G (U)·s ⊆ G (U) } for every open U ⊆ X, is, by deﬁnition, an OX -algebra. Clearly, it is coherent and has A as its stalk at the generic point. It is, therefore, an order.

7.3. Remarks. i) If X = Spec R is the spectrum of a Noetherian, integrally closed domain, then every global section s ∈ Γ(X, A ) ⊆ A of every order A is integral over R. ii) On a central simple algebra A over a ﬁeld K,thereisthereduced trace tr: A → K. It admits the property that (x, y) → tr(xy) is a non-degenerate bilinear form [Re, Theorem (9.26)]. Further, if R ⊆ K is integrally closed and s is a section of an order over Spec R, then tr(s) ∈ R.

7.4. Lemma. Let X be an integral scheme that is Noetherian and normal. Then every order A over X in a central simple algebra A over Q(X) is contained in a maximal order.

Proof. Let Spec R ⊆ X be affine open. Then A |Spec R is generated by finitely many global sections s1,...,sl. Clearly, s1,...,sl generate A over K. If A ⊃ A |Spec R is any order and t is any global section, then ts1, ... ,tsl are global sections and tr(ts1),...,tr(tsl) ∈ R. This list of conditions defines a finite R- module, i.e., a coherent sheaf A over Spec R, containing every order that contains A.

104 on the brauer group of a scheme [Chap. III

Since X is covered by finitely many open affine sets, every ascending chain of orders in A over X stops after finitely many steps.

7.5. Lemma. Let X be an integral scheme that is Noetherian and normal. Let A be a maximal order over X in a central simple algebra A over Q(X). ∈ A | Then, for every (not necessarily closed) point x X, Spec OX,x is a maximal order in A over Spec OX,x. Proof. For X aﬃne, this follows from [Re, Theorem (1.11)].

For the general case, it remains to verify that A |V is maximal for every aﬃne open subscheme Spec R = V ⊆ X. We shall do this by contradiction. Assume there would be a larger order B over V . Then there exists some non- zero r ∈ R such that r·B ⊆ A .Extendr·B to a coherent subsheaf E ⊆ A on X in the obvious way. I.e., a local section s ∈ E (U) is,bydeﬁnition,asectionofA such that s|U∩V ∈ (r·B)(U ∩ V ). Now, consider the order (E : E ) over X. Its restriction to V is easily understood. One has (E : E )|V =((r ·B):(r ·B)) = (B : B)=B as r commutes with B and B is an order. In particular, A |V ⊂ B has the property that E |V ·A |V ⊆ E |V . Every local section s ∈ (E · A )(U) is, therefore, a local section of A with the additional property that s|U∩V ∈ E (U ∩ V ). By construction of E , this means s ∈ E (U). Hence, E ·A ⊆ E and A ⊆ (E : E ).SinceA is a maximal order, this implies A =(E : E ).

Together with (E : E )|V = B, this shows A |V = B, in contradiction to our assumption that B would be a larger order. (This is nothing but a geometrization of the proof for [Re, Theorem (1.11)].)

Our interest in maximal orders comes from the following proposition.

7.6. Proposition. Let X be a Noetherian, normal, integral scheme with generic point η,andletA be an Azumaya algebra over X.

Then A is a maximal order in Aη over X. Proof. It is clear from the deﬁnition that A is an order. If it were not maximal, then there would be a larger order B. B admits a local section s ∈ A (U) over a certain subscheme U. We may therefore assume that X = Spec R is aﬃne, A = A0,andB = B0 for B A. ⊗ op ⊗ op We have A R A =Mn(R) for a certain n ∈ Æ.ThenO := B R A Mn(R) would be a larger order. I.e., the order Mn(R) would not be maximal.

Choose M ∈ O \ Mn(R). There is one entry x not in R. Elementary matrix operations produce the diagonal matrix ⎛ ⎞ x 0 ··· 0 ⎜ ⎟ ⎜ 0 x ··· 0 ⎟ ⎜ . . . . ⎟ ∈ O. ⎝ . . .. . ⎠ 00··· x This element alone generates the commutative R-algebra R[x], which cannot be ﬁnite since R is integrally closed. This is a contradiction.

Sec. 7] the theorem of auslander and goldman 105

7.7. Remark. The same approach yields that Mn(A ) is a maximal order as soon as A is. ii. The theorem of Auslander and Goldman.

Maximal orders over discrete valuation rings enjoy particular properties.

7.8. Proposition. Let R be a discrete valuation ring, let K = Q(R),and let A be an Azumaya algebra over K. Then all maximal orders in A over R are conjugate to each other. Proof. This is [Re, Theorem (18.7).iii)].

7.9. Theorem (M. Auslander and O. Goldman). Let X be an integral scheme that is regular, separated, and quasi-compact. Assume that the dimension of X is at most two. i) Then one has 2 Br(X)= Br(Spec OX,x) , codim x=1 where the intersection takes place in Br(Q(X)). ii) In particular, Br(X)=Br (X). Proof. Recall that, by Corollary 6.5, Br(Spec OX,x)=Br (Spec OX,x).Fur- ther, Br(Q(X)) = Br(Q(X)). By virtue of the general Fact 5.2, we have that Br(X) ⊆ Br(Q(X)) and Br(Spec OX,x) ⊆ Br(Q(X)). Thus, the natural maps Br(X) → Br(Spec OX,x) are all injective. This shows the “⊆”-part of i). The same argument is true for the cohomological Brauer group. We obtain that 2 Br (X) ⊆ Br(Spec OX,x) . (‡) codim x=1 This, in turn, makes sure that the second assertion immediately follows from the ﬁrst. Indeed, the inclusion (‡) together with i) immediately yields Br(X) ⊆ Br(X). The inclusion the other way round was established in Fact 5.2. The main part is to prove “⊇”. For that, let α ∈ Br(Q(X)) such that 2 α ∈ Br(Spec OX,x) . codim x=1 We choose an Azumaya algebra A over Q(X) corresponding to the class α. By Lemma 7.4 there exists a maximal OX -order A in A. We claim A is a locally free sheaf.

This may be tested locally. Let x ∈ X be any point on X.WewriteR := OX,x and denote the maximal ideal in R by m. The maximality property of A implies A A A ∨∨ that is equal to its bidual. In particular, x = x . We consider the short exact sequence

−→ −→ −→ A ∨ −→ 0 M F0 x 0

106 on the brauer group of a scheme [Chap. III of OX,x-modules with F0 free. Dualizing yields /A ∨∨ / ∨ / ∨ / 0 x F0 M ... .

A ∨ ∨ ∨ M := HomR(M,R) is a torsion-free R-module. Thus, M → M ⊗RK is injective. We see M ∨ may be embedded into a K-vector space. It, therefore, allows an embedding into a free R-module F1,too,

−→ A −→ ∨ −→ −→ −→ 0 x F0 F1 Q 0 .

R is of cohomological dimension ≤ 2 [Mat, Theorem 19.2]. To establish that Ax is R A free, it is suﬃcient to verify that Tori ( x,R/m)=0for i =1and i =2[Mat, §19, Lemma 1].

For this, writing Q1 := ker(F1 → Q),wesee

R A R Tor2 ( x,R/m)=Tor3 (Q1,R/m)=0 and R A R R Tor1 ( x,R/m)=Tor2 (Q1,R/m)=Tor3 (Q, R/m)=0.

A is a locally free sheaf. Corollary 2.5 shows that the non-Azumaya locus of A is a closed subset T ⊆ X, pure of codimension one. We have to show that T = ∅. Assume the contrary. Then there is an irreducible divisor D = {x}⊆T ,forx a codimension one point. By assumption, A allows an extension to Spec OX,x as an B A | Azumaya algebra . Furthermore, we may consider Spec OX,x . Thisisamaximal O B A | order in A over Spec X,x. As maximal orders, and Spec OX,x are conjugate A | by Proposition 7.8. In particular, Spec OX,x is an Azumaya algebra. Thus, the assumption D = {x}⊆T leads to a contradiction.

7.10. Remark. Using more refined methods, it was shown that the two Brauer groups coincide in several more cases. i) The most general result in this direction is due to O. Gabber. It shows that Br(X)=Br (X)tors when X allows an ample invertible sheaf. It seems that Gab- ber’s original proof is not publicly available, but there is a different proof due to A. J. de Jong [dJo2]. For the particular case that X = U ∪ V is the union of two affine schemes such that U ∩ V is again affine, see [Ga, Hoo]or[K/O80]. ii) Quite recently, S. Schröer [Schr]showedBr(X)=Br (X)tors for any quasi- compact, separated, geometrically normal algebraic surface. If one assumes X to be regular and separated, then the subscript may be omitted in view of Fact 5.2.

7.11. Remarks. a) A. Grothendieck knew that Br (X) may be non-torsion,

even for an aﬃne normal surface over , in 1965/66 already [GrBrII, Remarque 1.11.b].

Sec. 8] examples 107 b) D. Edidin, B. Hassett, A. Kresch, and A. Vistoli [E/H/K/V](seealso[Bt])

constructed an example of a non-separated surface such that Br (X)= /2 , but Br(X)=0. X is the union of two aﬃne schemes, regular in codimension one, but not normal.

8. Examples i. Some rings and ﬁelds from number theory.

8.1. Examples. Let p be a prime, and let K be a finite field extension of Ép. i) Then Br(Spec K)=É/ . This is shown by local class field theory [Se67, Sec. 1, Theorems 1 and 2]. Classi- cally, the isomorphism is given as follows. It turns out that Br(Spec K)=H2 Gal(Knr/K), (Knr)∗

nr nr for K the maximal unramiﬁed extension. The valuation ν : K → induces ∼

2 nr nr ∗ = 2 nr 2

H Gal(K /K), (K ) −→ H Gal(K /K), = H ( , )

É É = Hom( , / )= / . ii) For OK , the ring of integers, one has Br(Spec OK )=0. Indeed, in the description of Br(Spec K)=Br(Spec K) from Proposition 5.8, the canonical homomorphism

 É Br (Spec K) −→ Hom Gal( p/ p), / is bijective.

8.2. Examples. i) One has Br(Spec Ê)= 2 / .

ii) One has Br(Spec )=0.

8.3. Examples. Let K be a number ﬁeld. i) Then

⊕ −→ É Br(Spec K)=ker s: É/ 2 / / .

p prime σ : K→Ê Here, s is just the summation. This is shown by global class ﬁeld theory [Ta67, 11.2]. The isomorphism is induced by the canonical maps

iν : Br(Spec K) → Br(Spec Kν )

∈ for ν Val(K),composedwiththeinvariant map invν : Br(Spec Kν ) → É/ (or

1 to 2 / or 0, respectively) from Example 8.1.i).

108 on the brauer group of a scheme [Chap. III ii) Denote by r the number of real embeddings of K. Then, for the integer ring OK ,

one has 1 r−1

≥ ( / ) if r 1 , Br(Spec O )= 2 K 0otherwise.

In particular, Br(Spec )=0. Here, the canonical homomorphism

Br (Spec K) −→ Hom(Gx, É/ ) codim x=1 from Proposition 5.8 is nothing but the projection

1 −→ ⊕ −→ É É ker s: É/ 2 / / / .

p prime σ : K→Ê p prime ii. Geometric examples.

8.4. Lemma. Let n ∈ Æ and X be any scheme such that n is invertible on X. Then there is the short exact sequence

−→ −→c1 2 −→ −→ 0 Pic(X)/n Pic(X) Hét(X, μn) Br (X)n 0 .

Proof. This follows from the long exact sequence in étale cohomology associated to the Kummer sequence

−→ n −→ −→ 0 μn −→ m m 0 .

8.5. Proposition. Let k ⊆ be an algebraically closed ﬁeld, and let X be a scheme that is regular and proper over k.

−

rk H (X( ), ) rk NS(X) 3

⊕ Then Br (X)=(É/ ) H (X( ), )tors. Proof. Br(X) is torsion in view of Fact 5.2. For the middle term in the exact sequence above, by [SGA4, Exp. XVI, Corol- 2 ∼ 2 laire 1.6], we have Hét(X, μn) = Hét(X ,μn). Further, the comparison theorem

2 ∼ 2

[SGA4, Exp. XI, Théorème 4.4] shows Hét(X ,μn) = H (X( ), /n ). On the other hand, by the theorem of Murre and Oort [SGA6, Exp. XII, Corol- laires 1.2 et 1.5.a)], the Picard functor is representable by a scheme PicX/k . This is a group scheme, hence smooth over k. It is the disjoint union of quasi-

projective k-schemes. Among other properties, one has PicX/k (k)=Pic(X) and PicX/k ( )=Pic(X ).

The connected components of PicX/k are in bijection with NS(X). Base change 

from k to does not change these components. In particular, NS(X)=NS(X ). Corresponding to this, Pic(X) is an extension of the Néron–Severi group NS(X), which is ﬁnitely generated, by a divisible group,

0 −→ Pic0(X) −→ Pic(X) −→ NS(X) −→ 0 .

Sec. 8] examples 109

∼ ∼ This shows Pic(X)/n Pic(X) = NS(X)/nNS(X) = NS(X )/nNS(X ). By conse-

quence, we may assume from now on that k = . Doing this, we ﬁrst observe that the homomorphism

−→ 2 Pic(X)/n Pic(X) Hét(X, μn) induced by the Kummer sequence is compatible with the usual ﬁrst Chern class homomorphism coming from the exponential sequence. Indeed, there is the commutative diagram / 2πi /O exp /O∗ / 0 X X 0

2πi · 1 · exp( n ) exp( n ) = · n / /O∗ ( ) /O∗ / 0 μn X X 0 of exact sequences of sheaves, which induces the commutative diagram

= / 1 ∗ / 2 O Pic(X) H (X, X ) H (X, )

= = / 1 O∗ / 2 Pic(X) H (X, X ) H (X, μn) . By the Lefschetz theorem on (1, 1)-classes [G/H, p. 163], the cokernel of the ﬁrst Chern class homomorphism

2 c1 : Pic(X) −→ H (X, )

2 is always torsion-free. Writing r := rk H (X( ), ) and ρ := rk NS(X),wethere- ∼ r−ρ fore have coker c1 = . Thus, the cokernel of the induced homomorphism

⊗ → ⊗ c1 /n : Pic(X)/n Pic(X) H (X, ) /n

r−ρ is isomorphic to ( /n ) . Further, the universal coeﬃcient theorem [Sp, Chap. 5, Sec. 5, Theorem 10] yields the short exact sequence

2 2 3

⊗ −→ −→ → → 

0 H (X( ), ) /n H (X( ), /n ) H (X( ), )n 0 .

Thus, from Lemma 8.4 we see that there is a surjective canonical homo-

3 morphism Br (X)n → H (X( ), )n, the kernel of which is isomorphic to

∼ r−ρ

⊗ coker(c1 /n ) = ( /n ) . Altogether, there is a short exact sequence

−→ 0 −→ H −→ Br (X) −→ H (X( ), )tors 0,

r−ρ where #Hn = n for each n ∈ Æ. A simple application of the structure theorem

r−ρ for ﬁnite abelian groups shows that ( É/ ) is the only torsion group having this property.

r−ρ Finally, there are no non-trivial extensions of a ﬁnite torsion group by ( É/ ) .

110 on the brauer group of a scheme [Chap. III

8.6. Remark. According to the Lefschetz principle, this result allows a generalization to an arbitrary algebraically closed ﬁeld k of characteristic zero. Indeed, to describe the scheme X only ﬁnitely many data are needed. ∼ ×

Thus, X = X0 Spec k0 Spec k for a certain scheme X0 over a ﬁeld k0 that is ﬁnitely generated over É. k0 allows an embedding into .

8.7. Examples. Let k ⊆ be an algebraically closed ﬁeld. i) Let X be a smooth proper curve over k.ThenBr(X)=0. 1 Actually, in this case, one has even Br(Q(X)) = 0 by the Theorem of Tsen [SGA4 2 , Arcata, Sec. III, Théorème (2.3)]. ii) Let X be a rational surface over k that is proper, but not necessarily minimal. Then Br(X)=0.

22−rk Pic(X)

iii) Let X be a K3 surface over k.ThenBr(X)=(É/ ) . iv) Let X be an Enriques surface over k.ThenBr(X)= /2 .

8.8. Fact. Let k ⊆ be an algebraically closed ﬁeld, and let X be a smooth ≥ n complete intersection of dimension 3 in Pk . Then Br (X)=0.

Proof. The Lefschetz hyperplane theorem [Bot, Corollary of Theorem 1] implies

 that H1(X( ), )=0and H2(X( ), )= . From the universal-coeﬃcient theo-

rem for cohomology [Sp, Chap. 5, Sec. 5, Theorem 3], we deduce H (X( ), )= and

3 ∼

 H (X( ), ) = Hom(H3(X( ), ), ) .

In particular, H (X( ), ) is torsion-free. iii. Varieties over a number ﬁeld or local ﬁeld.

8.9. In this subsection, we deal with the case that X is a scheme over an algebraically non-closed field K. The relevant cases for us are that K is either a number field or a local field. It is not true that the homomorphism Br(Spec K) → Br(X), induced by the structural map, is always injective. For this, an additional hypothesis is needed.

8.10. Proposition. Let K be a field, and let π : X → Spec K be any scheme over K. Assume that a) either K is a local field and X(K) = ∅ ∅ b) or K is a number field and ν∈Val(K) X(Kν) = .

Then the canonical map π∗ : Br(Spec K) → Br(X) is injective.

Sec. 8] examples 111

Proof. a) A K-valued point x: Spec K → X induces a section of π∗. 1 b) Put S := ν Spec Kν . The assumption says that X has an S-valued point. Therefore, the natural homomorphism i: Br(K) → Br(S)= Br(Kν ) ν factors via π∗. i is injective by Example 8.3.i).

8.11. Proposition. Let K be a number ﬁeld or a local ﬁeld, and let π : X → Spec K be a geometrically integral scheme that is proper over K. i) Then there is an exact sequence

∗ −→ −→i Gal(K/K) −→ −→π 0 Pic(X) Pic(XK ) Br(Spec K) ∗ −→π →i −→ 1 −→ ker(Br (X) Br (XK )) H Gal(K/K), Pic(XK ) 0 . → Here, i and i are induced by the natural morphism XK X. ii) If π∗ : Br(Spec K) → Br (X) is injective, then there is the short exact sequence

∗ → →π →i → 1 → 0 Br(Spec K) ker(Br (X) Br (XK )) H Gal(K/K), Pic(XK ) 0 .

∼ Gal(K/K) Further, Pic(X) = Pic(XK ) .

Proof. Consider the Hochschild–Serre spectral sequence

p,q p q n ⇒ E2 := H Gal(K/K),Hét(XK , m) = Hét(X, m) in étale cohomology. 0 ∗ As X is integral and proper over K,wehavethatHét(XK , m)=K .Further,

1 2 Hét(XK , m)=Pic(XK ) and Hét(XK , m)=Br (XK ). 0,1 Gal(K/K) 2,0 This yields E2 = Pic(XK ) and E2 = Br(Spec K).Furthermore, E1,1 = H1 Gal(K/K), Pic(X ) and E0,2 = Br (X )Gal(K/K). 2 K 2 K 1,0 1 ∗ The classical Theorem Hilbert 90 shows that E2 = H Gal(K/K), K =0. In addition, if K is a number field, then one has 3,0 3 ∗ E2 = H Gal(K/K), K =0 by [Ta67 , Section 11.4]. Otherwise, if K is a local field, then, by class field the- 3 ∗ ∼ 1 ory, H Gal (L/K),L = H Gal(L/K), =0for every finite extension of K. Thus, H3 Gal(K/K), K∗ =0.

1 1 2 2 Finally, E = Hét(X, m)=Pic(X) and E = Hét(X, m)=Br (X). The sequence in i) is nothing but a sequence of lower order terms in the spectral sequence E. ii) directly follows from i).

112 on the brauer group of a scheme [Chap. III

8.12. Corollary. Let K be a number ﬁeld or local ﬁeld, and let π : X → Spec K be a geometrically integral scheme that is proper over K. i) Then Br(X) is a countable group.

2 2

 ii) Suppose that H (X( ), ) is torsion-free, rk NS(X)=rk H (X( ), ),and

1 ∗ rk H (X( ), )=0.ThenBr(X)/π Br(Spec K) is ﬁnite.

Proof. i) Br(XK ) is a countable group by Proposition 8.5. It is therefore suﬃcient → to verify that ker(Br(X) Br(XK )) is a countable group. We will even show that → ker(Br (X) Br (XK )) is countable. It is known from Example 8.3.i) that Br(Spec K) is countable. We are left with 1 proving H Gal(K/K), Pic(XK ) is a countable group. There is a short exact sequence of Gal(K/K)-modules

−→ 0 −→ −→ −→ 0 Pic (XK ) Pic(XK ) NS(XK ) 0 . 1 Here, H Gal(K/K), NS(XK ) is finite since NS(XK ) is a finitely generated abelian group acted upon by a finite quotient of Gal(K/K). 0 0 On the other hand, Pic (XK ) is divisible and all the groups Pic (XK )n are finite. The Kummer sequence

−→ 0 −→ −→n −→ 0 Pic (XK )n Pic(XK ) Pic(XK ) 0 1 0 → 1 0 induces a surjection H Gal(K/K), Pic (XK )n H Gal (K/K), Pic (XK ) n. 1 0 1 0 In particular, H Gal(K/K), Pic (XK ) n is ﬁnite and H Gal(K/K), Pic (XK ) is countable.

2 ii) Here, the assumption rk NS(X)=rk H (X( ), ) makes sure that Br(XK ) is ﬁnite. It is therefore suﬃcient to verify that

→ ∗ ker(Br(X) Br(XK ))/π Br(Spec K) is ﬁnite. Again, we will even show that → ∗ ∼ 1 ker(Br (X) Br (XK ))/π Br(Spec K) = H Gal(K/K), Pic(XK ) is ﬁnite.

1 For this, the assumption rk H (X( ), )=0implies that Pic(XK )=NS(XK ). As this is a torsion-free abelian group, the assertion follows.

8.13. Remark. ThecasethatBr(XK )=0is of particular interest. For example, this happens if XK is a rational surface or a smooth complete intersection of dimension ≥ 3 in Pn (cf. Subsection ii, Geometric examples, above). K In this case, the short exact sequence in assertion ii) of Proposition 8.11 simply means ∗ ∼ 1 Br (X)/π Br(Spec K) = H Gal(K/K), Pic(XK ) .

Sec. 8] examples 113

8.14. Fact. Let K be a number field or a local field, and let X be a smooth ≥ n complete intersection of dimension 3 in PK . i) Then Br(X)=Br (X). ii) The homomorphism π∗ : Br(Spec K) → Br(X) induced by the structural map is surjective. ∅ ∅ iii) If X(K) = for K a local field or ν∈Val(K) X(Kν ) = for K anumberfield, then Br(X)=Br(Spec K).

Proof. The Lefschetz hyperplane theorem implies that Pic(XK )= . Therefore, 1 1 H Gal(K/K), Pic(XK ) = H Gal(K/K), =0. ∗ → Since Br (XK )=0, Proposition 8.11 shows that π : Br(Spec K) Br (X) is surjective. In particular, π∗ does not factorize via any proper subgroup of Br (X). Hence, Br(X)=Br (X).Thisprovesi)andii). iii) follows now from Proposition 8.10.

8.15. In the situation of a curve over a local ﬁeld, Br(X)/π∗Br(Spec K) has been computed by S. Lichtenbaum. The following result is often referred to as Lichtenbaum duality.

8.16. Theorem (Lichtenbaum). Let p be a prime number, let K be a ﬁnite

extension of Ép,andletX be a geometrically integral curve that is proper and smooth over K. Then the canonical pairing

× −→ Br(X) X(K) É/ , → | (α, x) invνp (α x) ∼

∗ = 0 induces an isomorphism Br(X)/π Br(Spec K) −→ Hom(Pic (X), É/ ). Proof. This is [Lic, Corollary 1]. iv. Manin’s formula. 1 8.17. If XK is a rational surface, then H Gal(K/K), Pic(XK ) is often

eﬀectively computable using the following result due to Yu. I. Manin.

Ðº ÈÖÐÓ Ò º¿ 8.18. Proposition (Manin, [Man, IV, ]). Let X be a regular, integral scheme that is of finite type over a field k.Supposefur- sep ther, we are given a finite, G := Gal(k /k)-invariant set {Di} of divisors sep on X := X ×Spec k Spec k generating Pic(X).

Denote by S ⊆ Div(X) the group generated by {Di} and by S0 ⊆ S the subgroup of all principal divisors. Finally, let H ⊆ G be a normal subgroup acting trivially on {Di}.

114 on the brauer group of a scheme [Chap. III

Assume i) Pic(X) is a free abelian group, and ii) there is a perfect pairing

Pic(X) × Pic(X) −→ .

Then there is a canonical isomorphism

1 i H (G, Pic(X)) −→ Hom((NS ∩ S0)/N S0, É/ ) .

Here, N : S → S denotes the norm map on S as a G/H-module. Proof. By Lemma 8.19, we have

−1 ∼ Hˆ (G/H, Pic(X)) = (NS ∩ S0)/N S0 .

To this relation, we apply the duality theorem [C/E, Chap. XII, Corollary 6.5]. It shows that

ˆ 0 ∼ ˆ −1

É H (G/H, Hom(Pic(X), É/ )) = Hom(H (G/H, Pic(X)), / )

∩ = Hom((NS S0)/N S0, É/ ) .

It remains to construct a canonical isomorphism

ˆ 0 ∼ 1 H (G/H, Hom(Pic(X), É/ )) = H (G, Pic(X)) .

For this, since Pic(X) is free, we have a short exact sequence of G/H-modules

−→ −→ É −→ É −→ 0 Hom(Pic(X), ) Hom(Pic(X), ) Hom(Pic(X), / ) 0 .

As Hom(Pic(X), É) is uniquely divisible, this shows

ˆ 0 ∼ 1

H (G/H, Hom(Pic(X), É/ )) = H (G/H, Hom(Pic(X), )) .

We ﬁnally apply the perfect pairing on Pic(X).

8.19. Lemma. Let G be a finite group, and let D be a finite G-set. Con- sider the free abelian group S over D as a G-module. Then, for every G-submodule S0 ⊆ S, there is a canonical isomorphism ∼ −1 = Hˆ (G, P ) −→ (NS ∩ S0)/N S0 , where P := S/S0. Proof. We first calculate Hˆ −1(G, S) directly, according to the definition. Ob-

serve that S is a direct sum of G-modules of the form [G/H] for normal subgroups H ⊆ G. For these, one has

∼ H0(G, [G/H]) = ,

Sec. 8] examples 115 →

and the isomorphism is given by the map [G/H] sending a formal sum to → the sum of all coeﬃcients. Further, the homomorphism [G/H] mapping 1 ˆ −1 to g∈G/H g has no kernel. Thus, H (G, S)=0. The short exact sequence

0 −→ S0 −→ S −→ P −→ 0 of G-modules therefore induces a long exact sequence

−1 0 0 0 −→ Hˆ (G, P ) −→ Hˆ (G, S0) −→ Hˆ (G, S) on Tate cohomology. We may consequently write

ˆ −1 G → G ∼ ∩ H (G, P )=ker(S0 /N S0 S /N S) = (NS S0)/N S0 .

8.20. Remark. The assumptions are fulﬁlled if X is a proper surface such

O that H1(X( ), )=0and H (X, X )=0. For example, X may be a (not necessarily minimal) rational surface or a K3 surface.

1 2

 The ﬁrst condition implies that H (X( ), )=0and that H (X( ), ) is torsion- free. In particular, the ﬁrst Chern class

→ 

c1 : Pic(X ) H (X( ), )

2 is injective. The second condition makes sure it is a surjection. As H (X( ), ) is torsion-free, Poincaré duality yields that the intersection pairing is perfect.

8.21. Let us consider the case that X is a smooth cubic surface in a bit more detail. It is well known that, on a cubic surface X over an algebraically closed ﬁeld, there ∼ 7 are exactly 27 lines D1,...,D27.Further,Pic(X ) = is generated by the classes of these lines.

The set L := {D1, ... ,D27} is equipped with the intersection product

 , : L ×L →{−1, 0, 1}. The pair (L , , ) is the same for all smooth cubic

Ðº ÌÓÖÑ ½º ºµ surfaces. It is well known [Man, IV, ] that the group of permutations of L respecting , is isomorphic to W (E6). We ﬁx such an isomorphism. ∼ 27 L Further, we put S = to be the free group over and S0 := aiDi aiDi,Dj =0forj =1,...,27 . i i Let X be a smooth cubic surface over a number ﬁeld K.ThenGal(K/K) L operates canonically∼ on the set X of the 27 lines on XK . Fix a bijection = iX : LX −→ L respecting the intersection pairing. This induces a group homomorphism ιX : Gal(K/K) → W (E6). We denote its image by G ⊆ W (E6).

As LX is clearly Gal(K/K)-invariant, we may apply Proposition 8.18. It shows

1 1

∩ É H Gal(K/K), Pic(XK ) = H (G, S/S0)=Hom (NS S0)/N S0, / .

116 on the brauer group of a scheme [Chap. III

The right-hand side depends only on the conjugacy class of the subgroup G ⊆ W (E6). Actually, it depends only on the decomposition of L into G-orbits.

8.22. Remark. Almost as a byproduct, we may write down a similar formula for another arithmetic invariant of the cubic surface X, namely for the Pi- card rank. Indeed, Proposition 8.11.i) shows 0 rk Pic(X)=rk H Gal(K/K), Pic(XK ) ,

G and, therefore, rk Pic(X)=rk(S/S0) = rk N(S/S0)=rk[(NS + S0)/S0]. I.e.,

rk Pic(X)=rk NS/(NS ∩ S0)

= rk NS − rk(NS ∩ S0) .

Observe that rk NS is the number of Gal(K/K) orbits, into which the 27 lines ∩ are decomposed. The group NS S0 has to be computed anyway for 1 H Gal(K/K), Pic(XK ) .

8.23. Explicit computation. There are exactly 350 conjugacy classes of subgroups in W (E6).UsingGAP, we computed the right-hand side in each case. The result is in Table 1.

Table 1. H1(G, Pic) and rk Pic(X) for smooth cubic surfaces

......

347 #U = 1440 [ 2, 2 ], #H^1 = 2, Rk(Pic) = 1, Orbits [ 12, 15 ] 348 #U = 1920 [ 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 10, 16 ] 349 #U = 25920 [ ], transitive 350 #U = 51840 [ 2 ], transitive ThecodewritteninGAP as well as the full list are reproduced in the Appendix. 1 8.24. It turns out that H Gal(K/K), Pic(XK ) is isomorphic to

0 for 257 classes,

/2 for 65 classes, /3 for 16 classes,

2 ( /2 ) for 11 classes,

2 ( /3 ) for one class. The computations took approximately 28 seconds of CPU time. 1 In 25 of the 257 classes, we have H Gal(K/K), Pic(XK ) =0for the trivial reason that the operation of G on the 27 lines is transitive. Among these classes, there are the general cubic surfaces. I.e., the case that G = W (E ). 6 1 A diﬀerent case where H Gal(K/K), Pic(XK ) =0is that when G =0, this means that all 27 lines are deﬁned over K.

Sec. 8] examples 117

1 2 ThecasewhereH Gal(K/K), Pic(X ) =(/3 ) was described already by

K ∼

Ðº ËÐ×ØÚ º Yu.I.Manin[Man, VI, ]. It occurs in a situation when G = /3 splits the 27 lines into nine orbits of three lines each. This is realized, for√ example, 3 3 3 3 3 3 by the diagonal cubic surface “x + y + z + dw =0”inPK when K( d)/K is a cubic Galois extension.

8.25. Remark. That only these five groups may appear was known to Sir Peter Swinnerton-Dyer in 1993 already. Swinnerton-Dyer’s proof fills almost the entire article [SD93]. A different proof is given in the Ph.D. thesis of P. K. Corn [Cor]. That proof consists of a mixture of mathematical arguments with computer work. A purely computational approach was, seemingly, considered rather hopeless at that time (cf. [Cor, Proposition 1.3.11] and the remarks before).

8.26. Remark. We ﬁnd a Picard rank of 1 for 137 classes, 2 for 133 classes, 3 for 62 classes, 4 for twelve classes, 5 for four classes, 6 for one class, 7 for one class.

8.27. Remark. In the next chapter, we will use the list given in Table 1 in order to describe the Brauer–Manin obstruction for a large class of diagonal cubic surfaces.

CHAPTER IV

An application: the Brauer–Manin obstruction

I was at ﬁrst almost frightened when I saw such mathematical force made to bear upon the subject, and then wondered to see that the subject stood it so well. Michael Faraday (1857, in a letter to J. C. Maxwell)

1. Adelic points i. The concept of an adelic point.

1.1. Deﬁnition. Let K be a number ﬁeld, and let X be a scheme over K. → Then an adelic point on X is a morphism Spec K X of K-schemes from the spectrum of the adele ring [Cas67, Sec. 14]. The set of all adelic points on X is

denoted by X( K ).

1.2. Remark. The scheme Spec K is not Noetherian.

1.3. Remark. K is a subring of K via the diagonal homomorphism. Thus, every K-valued point on X induces an adelic point.

1.4. Remarks. i) K is canonically a sub-K-algebra of ν Kν .There- fore, every adelic point on X gives rise to a K -valued point. ν ν ii) A K -valued point induces a K -valued point for every ν ∈ Val(K). ν ν ν If the scheme X is separated and quasi-compact, then the ν Kν -valued points are in a canonical bijection with ν X(Kν ). This is a technical point, which we defer to Lemma 3.2.

1.5. Lemma. Let K be a number ﬁeld, and let X be a separated K-scheme. ∈ Then the sequence x =(xν )ν ν X(Kν ) determines the corresponding adelic point, if it exists, uniquely. Proof. We ﬁrst observe that the set 3

Spec Kν ⊂ Spec K ν∈Val(K) is Zariski dense. Indeed, a base of the open subsets is provided by the sets D(f) for 0 = f =(... ,fν , ...) ∈ K .Iffν =0,thenSpecKν is contained in D(f). → Now suppose that two morphisms x1,x2 : Spec K X induce the same element in ν X(Kν ). This means simply that x1 and x2 coincide on ν∈Val(K) Spec Kν .

119

120 an application: the brauer–manin obstruction [Chap. IV

In the category of schemes, there exist a universal object V and a morphism → ◦ ◦ i: V Spec K such that x1 i = x2 i. The scheme V could be called the difference kernel of x1 and x2. Accordingto[EGA, Chapitre I, Proposition (5.2.5)], i is a closed embedding. Therefore, x1 and x2 coincide on the Zariski closure of ν∈Val(K) Spec Kν .Thisis the whole of Spec K .

1.6. Lemma. Let K be a number field, let OK be its ring of integers, and let X be a separated O -scheme of finite type. Denote by X its generic fiber. K ∈ Then a ν Kν -valued point x =(xν )ν ν X(Kν ) is induced by an adelic point if O X and only if all but finitely many of the xν extend to Kν -valued points on . Proof. “⇐=” We are given a finite set S of valuations, including all Archimedean O → X ∈ → X ∈ ones, and morphisms Spec Kν for ν S and Spec Kν for ν S. ∈ O ∈ To ease notation, we write Rν := Kν for ν S and Rν := Kν for ν S. → X Thus, we have morphisms Spec Rν . We claim that these may be put together → X to give a morphism Spec R for R := ν∈Val(K) Rν . The assertion then follows → as there is a canonical morphism Spec K Spec R. X For that, we first note that the assertion is true when = Spec A is an → affine scheme. Indeed, to give a ring homomorphism A ν∈Val(K) Rν is the same as giving a system of ring homomorphisms {A → Rν }ν∈Val(K). ∼ We cover X by finitely many affine schemes Xj = Spec Aj, j ∈{1,...,n}.Given a system of morphisms {Spec Rν → X }ν∈Val(K), we may find a decomposition

Val(K)=S1 ∪ ... ∪ Sn into mutually disjoint subsets such that Spec Rν maps to Xj if ν ∈ Sj . For this, observethatalltheringsRν are local.

Now we use the assumption that the Xj are aﬃne. For j ﬁxed, the morphisms Spec Rν → Xj for ν ∈ Sj give rise to a morphism Spec Rν → Xj ⊆ X .

ν∈Sj Finally, observe that n ∼ Spec Rν = Spec Rν

j=1 ν∈Sj ν∈Val(K) since the index set is ﬁnite.

“=⇒”CoverX by aﬃne schemes X1,...,Xn,andwrite

X O j = Spec K [Tj1,...,Tjlj ]/Ij .

The adelic point given induces homomorphisms of OK -algebras O −→ ϕj : K [Tj1,...,Tjlj ]/Ij ( K )fj ,

Sec. 1] adelic points 121 where (f1, ... ,fn)=(1). Fix adeles g1, ... ,gn such that g1f1 + ... + gnfn =1, and write hjk ejk := ϕj (Tjk) fj ≥ where hjk ∈ K and ejk 0. We let S be the set of all valuations ν ∈ Val(K) that are either Archimedean or such that not all of the adeles gj and hjk are integral at ν. This is a ﬁnite set of valuations.

Assume ν ∈ S. Then, for one adele fi, we certainly have that (fi)ν ν ≥ 1. ∈ O Since (hjk)ν is integral, we see that [ϕj (Tjk)]ν Kν for every k. ϕj induces a morphism Spec OK → Xj as claimed.

1.7. Definition (The topology on X( K )). Let K be a number field, and let X be a separated K-scheme of finite type. Then X allows a model X that is separated and of finite type over the integer ring OK .ThesetsX(Kν ) carry natural topologies as ν-adic analytic spaces. X O ⊆ For ν non-Archimedean we have the subsets ( Kν ) X(Kν ) carrying the subspace topology.

We equip the set X( K ) of all adelic points on X with the restricted topological product topology [Cas67, Sec. 13]. A basis is provided by all direct products Γν ν ⊆ X O where Γν X(Kν ) is open for all ν and Γν = ( Kν ) for almost all ν.

1.8. Remarks. a) For two diﬀerent models X1 and X2, the isomorphism ∼ ∼

X −→= −→= X É ( 1) É X ( 2) may be extended to an open neighbourhood of the generic fiber. This implies X O ∼ X O 1( Kν ) = 2( Kν ) for all but finitely many valuations ν. Hence, the definition is independent on the choice of the model. The existence of a model is provided by Lemma 3.4.a).

b) X( K) is a Hausdorﬀ topological space satisfying the second axiom of count- ability.

1 1 i) If X = A , then there is a bijection between A ( K ) and the adele ring K .

1 The topology we deﬁned on A ( K ) coincides with the adele topology on K deﬁned in [Cas67, Sec. 14]. 1 ii) If X = A \{0},thenX( K ) corresponds to the idele group. It is, however, equipped with the restriction of the adele topology, not with the idele topology.

1.9. Lemma. Let K be a number ﬁeld, and let X be a scheme proper over K.Then

X( K )= X(Kν ) , ν∈Val(K) equipped with the Tychonov topology.

122 an application: the brauer–manin obstruction [Chap. IV

Proof. Lemma 3.4.b) guarantees the existence of a model X that is proper over OK except for a ﬁnite set S of special ﬁbers. The valuative criterion for properness X O ∈ implies that ( Kν )=X(Kν ) for ν S. ii. Weak approximation and the Hasse principle.

1.10. Definition. Let X be a separated scheme that is of finite type over a number field K. ∅ a) We say that X satisfies the Hasse principle if the condition X( K ) = implies X(K) = ∅.

b) X is said to satisfy weak approximation if X(K) is dense in X( K ).

1.11. Theorem. a) The following classes of projective schemes satisfy the Hasse principle: i) smooth projective quadrics, ii) Brauer–Severi varieties,

iii) smooth projective cubics in Pn for n ≥ 9, É n ≥ iv) smooth complete intersections of two quadrics in PK for n 8. b) The following projective schemes satisfy weak approximation: i) projective varieties rational over K, n ≥ ii) smooth complete intersections X of two quadrics in PK for n 6 satisfying X(K) = ∅.

1.12. Remark. These lists are not meant to be exhaustive. Further notes and references to the literature may be found in [Sk, Sec. 5.1].

2. The Brauer–Manin obstruction

2.1. Weak approximation and even the Hasse principle are not always satisfied. Isolated counterexamples have been known for a long time. Genus one curves violating the Hasse principle have been constructed by C.-E. Lind [Lin] as early as 1940 and by E. S. Selmer [Sel] in 1951. The first example of a cubic surface not fulfilling the Hasse principle is due to Sir Peter Swinnerton-Dyer [SD62]. A series of examples generalizing Swinner- ton-Dyer’s work has been given by L. J. Mordell [Mord]. We will generalize Mordell’s examples even further in Section 5. Diagonal cubic surfaces that are counterexamples to the Hasse principle have been constructed byJ.W.S. Cassels and M.J.T. Guy [Ca/G]aswellasA.Bremner [Bre]. These investigations were systematized by J.-L. Colliot-Thélène, D. Kanev- sky, and J. J. Sansuc [CT/K/S]. We will discuss this topic in more detail in Sec- tion 6. A singular quartic surface that is a counterexample to the Hasse principle has been given by V. A. Iskovskikh [Is].

Sec. 2] the brauer–manin obstruction 123

A method to explain all these examples in a uniﬁed manner was provided by Yu. I. Manin in his book on cubic forms [Man]. This is what is nowadays called the Brauer–Manin obstruction.

2.2. Definition. Let K be a number field, and let X be a scheme separated and of finite type over K.

i) Then, for each ν ∈ Val(K),thereisthelocal evaluation map, given by evν : Br(X) × X(Kν ) −→ É/ ,

(α, x) → invν (α|x) . ii) Further, there is the global evaluation map or Manin map

× −→ É ev: Br(X) X( ) / , K α, (xν)ν → evν (α, xν) . ν 2.3. Proposition. Let K be a number field, and let π : X → Spec K be a scheme that is separated and of finite type over K. Then a) the local evaluation map is i) additive in the first variable and ii) continuous in the second variable; ∗ iii) if α = π a for a ∈ Br(Spec K);thenevν (α, ·)=invν (a) is constant; iv) for each α ∈ Br(X), there exists a finite set S of valuations including all Archi- medean ones such that evν (α, x)=0 ∈ ∈ X O whenever ν S and x ( Kν ). b) The Manin map is well defined. Further, it is i) additive in the first variable and ii) continuous in the second variable; iii) in the first variable, ev factors via Br(X)/π∗Br(Spec K); iv) further, ev(α, x)=0

if x ∈ X(K) ⊆ X( K ). Proof. a) i) and iii) are obvious. ii) Let α ∈ Br(X) and x ∈ X(Kν ).Wehavetoshowthatevν (α, ·) is constant in a neighbourhood of x in the ν-adic topology.

By iii), we may assume that evν (α, x)=0. Then there is an Azumaya algebra A ∼ over X such that [A ]=α and A |x = Mn(Kν ). By Corollary III.6.4, this implies

∼ h A | Oh = Mn(O ) Spec X ,x XKν ,x Kν

124 an application: the brauer–manin obstruction [Chap. IV

∈ for the Henselization of the local ring at x XKν . To define this isomorphism, only finitely many data are required. Thus, there exist an étale morphism f : X → X ∼ and a Kν -valued point x ∈ X such that f(x )=x and A |X = Mn(OX ). The morphism f induces a local isomorphism of ν-adic analytic spaces. In partic- ∼ ular, A |y = Mn(Kν ) for y in a ν-adic neighbourhood of x. iv) By Lemma 3.4.a), there exists a model X of X that is separated and of finite type over OK . We may assume that X is reduced. Let A be an Azumaya algebra over X such that [A ]=α. By Lemma 3.5, there is an open subset X ◦ ⊆ X such that A may be extended to an Azumaya algebra A over X ◦. The closed subset X \X ◦ ⊂ X does not meet the generic fiber. Since X is of finite type, X \X ◦ is contained in finitely many special fibers. We put S to be the set of the corresponding valuations together with all Archimedean valuations.

We have to prove that evν (α, x)=0for ν ∈ S and x ∈ X(Kν ).Thepointx O → X O ∈ induces a morphism x: Spec Kν of K -schemes. The assumption ν S implies that x actually maps to X ◦. Over X ◦, there is the Azumaya algebra A extending A . The restriction of [A ] ∈ Br(X) along x: Spec Kν → X coincides with the restriction of ◦ x ⊂ [A ] ∈ Br(X ) along Spec Kν −→ X −→ X . The latter factorizes via O O Br(Spec Kν ). We note, finally, that Br(Spec Kν )=0by virtue of Example III.8.1.ii). b) In order to show that ev is well defined, one has to verify that the sum is always finite. This, in turn, immediately follows from a.iv). i) and iii) are direct consequences from a.i) and a.iii). ii) This follows from a.ii) together with a.iv). iv) is a consequence of the description of Br(K) given in Example III.8.3.i).

2.4. Remark. Let x ∈ X( K ) be an adelic point. If there exists a Brauer class α ∈ Br(X) such that ev(α, x) =0 ,thenx cannot be approximated by a sequence of K-valued points. The Brauer class α therefore “obstructs” the adelic point x from being approximated by rational points. This justiﬁes the name Brauer– Manin obstruction.

2.5. Notation. Let K be a number ﬁeld, and let π : X → Spec K be a scheme separated and of ﬁnite type over K. For α ∈ Br(X),wewrite

{ ∈ | } X( K) := x X( K ) ev(α, x)=0 .

We deﬁne 2

Br α X( K ) := X( K ) . α∈Br(X) Br 2.6. Remarks. a) According to Proposition 2.3.b.ii), X( K) is always a

closed subset of X( K ). Br b) Proposition 2.3.b.iv) shows X(K) ⊆ X( K ) .

Sec. 2] the brauer–manin obstruction 125

2.7. Remark. As shown in Corollary III.8.12.ii), there are many particular cases, in which Br(X)/π∗Br(Spec K) is actually a ﬁnite group.

In this case, one might choose a ﬁnite system α1, ... ,αn ∈ Br(X) generating

∗ Br α1 αn

∩ ∩ Br(X)/π Br(Spec K). This leads to X( K ) = X( K ) ... X( K) . By Proposition 2.3.a.iv), only ﬁnitely many places of K are involved in the evaluation of α1,...,αn.

2.8. Definition. Let K be a number field, and let π : X → Spec K be a scheme separated and of finite type over K.

Br a) If X( K ) = X( K ), then one says that, on X,thereisaBrauer–Manin obstruction to weak approximation.

∅ ∅ b) If X( K ) = and X( K) = , then one says that, on X,thereisaBrauer– Manin obstruction to rational points on X or that there is a Brauer–Manin obstruction to the Hasse principle on X.

Br ∅ 2.9. Remark. In the case that X( K ) = , many authors use the somewhat confusing formulation that the Brauer–Manin obstruction would be empty.

2.10. Remark. There is clearly no Brauer–Manin obstruction when Br(X)/π∗Br(Spec K)=0, neither to the Hasse principle nor to weak approximation. We know several such cases from the computations described above. For example, as indicated in III.8.23, we have Br(X)/π∗Br(Spec K)=0for X a general cubic surface or a smooth cubic surface such that all 27 lines are deﬁned over K. By Fact III.8.14, we have Br(X)/π∗Br(Spec K)=0when X is a smooth complete ≥ n intersection of dimension 3 in PK .

2.11. Deﬁnition. If the statement

Br ∅ ⇒ ∅ X( K) = = X(K) = is true for a certain class of separated K-schemes of ﬁnite type, then one says that the Brauer–Manin obstruction is the only obstruction to the Hasse principle for that class.

2.12. Conjecture ([Colliot-Thélène, cf. [CT/S81, Conjecture C]). The Brauer–Manin obstruction is the only obstruction to the Hasse principle for smooth cubic surfaces.

2.13. Remark. It is known that there are counterexamples when more general classes of varieties are allowed. For this, the reader should consult the article [Poo] of B. Poonen.

126 an application: the brauer–manin obstruction [Chap. IV

3. Technical lemmata

3.1. Let (Ki)i∈Æ be a sequence of ﬁelds. Then there is the canonical morphism of schemes

ι: Spec Ki −→ Spec Ki . ∈Æ i∈Æ i 3.2. Lemma. Let X be a scheme that is separated and quasi-compact. Then ι induces a bijection

X Ki −→ X(Ki) . ∈Æ i∈Æ i Proof. We first note that the assertion is true when X = Spec A is an affine scheme. Indeed, to give a ring homomorphism A → Ki is the same as giving a system i∈Æ → of ring homomorphisms (A Ki)i∈Æ . ∼ Surjectivity. We cover X by finitely many affine schemes Xj = Spec Aj, ∈{ } → j 1, ... ,n . Given a system of morphisms (Spec Ki X)i∈Æ , we may find a decomposition ∪ ∪ Æ = S1 ... Sn into mutually disjoint subsets such that Spec Ki maps to Xj if i ∈ Sj .

Now we use the assumption that the Xj are aﬃne. For j ﬁxed, the morphisms Spec Ki → Xj for i ∈ Sj give rise to a morphism Spec Ki → Xj ⊆ X.

i∈Sj Finally, observe that n ∼ Spec Ki = Spec Ki

j=1 i∈Sj i∈Æ since the index set is ﬁnite. Injectivity. Suppose two morphisms g1,g2 : Spec( Ki) → X induce the same 1 i∈Æ morphism on Spec Ki. Proposition (5.2.5) of [EGA, Chapitre I] implies that i∈Æ 1 g1 and g2 coincide on the Zariski closure of Spec Ki. 1 i∈Æ

We claim that Spec Ki is actually dense in Spec( Ki). Indeed, a base of ∈Æ i∈Æ i the open subsets is provided by the sets D(f) for 0 = f =(... ,fν , ...) ∈ Ki .

i∈Æ

If fν =0 ,thenSpecKν is contained in D(f). The claim follows.

3.3. Remark. The assertion of the lemma is certainly not true in general for1X an arbitrary scheme. Indeed, Yoneda’s lemma would then imply

that ι: Spec Ki → Spec( Ki) is an isomorphism. ∈Æ i∈Æ i ι induces, however, not even a bijection on closed points. In fact, there is the

ideal Ki ⊂ Ki. According to the Zorn’s Lemma, this ideal is contained ∈Æ i∈Æ i 1 in a maximal ideal that is diﬀerent from those in Spec Ki. i∈Æ

Sec. 3] technical lemmata 127

3.4. Lemma. Let K be a number field, and let X be a scheme that is separated and of finite type over K. a) Then there exists a model X of X. This means that X is a scheme that is O X × ∼ separated and of finite type over K and fulfills Spec OK Spec K = X. b) If X is proper over K,thenX is proper over Spec(OK )f for some 0 = f ∈ OK . Proof. First step. Existence of a model.

X is given by gluing together finitely many affine K-schemes Ui = Spec Ri.Thein- tersections Ui ∩ Uj are affine. Thus, gluing takes place along isomorphisms

−→ ϕij : Spec(Ri)gij Spec(Rj )gji .

The K-algebras may be described by generators and relations in the form

Ri = K[Ti1, ... ,Tiji ]/(hi1, ... ,hiki ). In the polynomials hik,gij, only ﬁnitely R ⊗ many denominators are occurring. We, therefore, have Ri = i OK K for some O R ∈ R ﬁnitely generated K -algebras i such that gji i. # R ⊗ → R ⊗ The ϕij induce isomorphisms ϕij :( j )gji OK K ( i)gij OK K.Forsome ∈ O # suitable 0 = fij K , the isomorphism ϕij is the base extension of an isomorphism (R ) ⊗O (O ) → (R ) ⊗O (O ) . j gji K K fij i gij K K fij R R Putting i := ( i)f for f := i,j fij,theϕij therefore extend to isomorphisms 0 R −→ R ϕij : Spec( i)gij Spec( j )gji .

The cocycle relations “ϕ0ik = ϕ0jk ◦ ϕ0ij” are satisfied after restriction to the R generic fiber. Since the schemes Spec( i)gij are of finite type, this implies that these relations are still fulfilled when the Ri are localized only by some suitable 0 = f ∈ OK .

We constructed gluing data for an OK -scheme X . X is clearly of finite type. Second step. Separatedness. ⊂ X × X Denote by Δ Spec OK the Zariski closure of the diagonal. Then the diagonal morphism δ : X → Δ is an isomorphism on the generic fiber. Since X and Δ are OK -schemes of finite type, δ is an isomorphism outside finitely many special fibers. We delete the corresponding special fibers from X . Third step. Properness. Assume first that X is reduced. Then, by Chow’s lemma, we have a surjective map → ⊆ n X X X from a projective K-scheme X PK .Let be the Zariski closure of X in Pn equipped with the induced reduced structure. This is a model of X, OK which is proper over Spec OK . Let X be any model of X in the sense of the previous steps. Then there exists some 0 = f ∈ OK such that the morphism X → X extends to a mor- O X × O → X phism of ( K )f -schemes g : Spec OK Spec( K )f .[EGA, Chapitre II, Corollaire (5.4.3.ii)] shows that im g is proper over Spec(OK )f . This implies that ⊆ X × O im g Spec OK Spec( K )f is a closed subscheme containing the generic fiber. X × O O In particular, the closure of X in Spec OK Spec( K )f is proper over Spec( K )f .

128 an application: the brauer–manin obstruction [Chap. IV

Return to the general case. Let X be any model of X in the sense of the two steps above. We may assume that X is equal to the closure of X. ∈ O X × O Then there exists some 0 = f K such that red Spec OK Spec( K )f is proper over Spec(OK )f .Byvirtueof[EGA, Chapitre II, Corollaire (5.4.6)], this suﬃces X × O for Spec OK Spec( K )f being proper.

3.5. Lemma. Let K be a number field, let X be a reduced scheme that is separated and of finite type over the integer ring OK ,andletA be an Azumaya algebra over the generic fiber X. Then there exist an open subset X ◦ ⊆ X containing X and an Azumaya algebra A over X ◦ that extends A .

Proof. First step. Extending A to a coherent sheaf.

Let j : X → X be the embedding of the generic ﬁber. Then j∗A is a quasi-coherent sheaf on X . We claim, j∗A contains a coherent subsheaf F such that F |X = A . Assume that would not be the case. We construct an increasing sequence of subsheaves of j∗A recursively as follows.

Put F1 := 1 for the section 1 ∈ Γ(X ,j∗A ).HavingFn already constructed, we have, by our assumption, Fn|X A .WeobservethatX is a Noetherian scheme. Therefore, j∗A is the union of its coherent subsheaves [Ha77, Chap. II, Exer- cise 5.15.a)]. By consequence, there exists a coherent sheaf G ⊆ j∗A such that G |X ⊆ Fn|X . We put Fn+1 := Fn + G . F |

This means ( n X )n∈Æ is a strictly increasing sequence of coherent subsheaves of A . This is a contradiction. Second step. Extending A to a locally free sheaf. We may assume X to be connected. Then

F ⊗ ψ(x):=dimk(x) X OX k(x) is constant on the generic fiber X.Asψ is upper semicontinuous, there is an open subset X ⊆ X containing the generic fiber such that ψ is constant on X . By reducedness, this suffices for F |X being locally free. Third step. Extending A to an Azumaya algebra. j∗A carries a natural structure of a sheaf of OX -algebras.

F ⊂ j∗A contains the constant section 1 by the construction given in the first step. To obtain a sheaf of algebras, we must establish closedness under multiplication. The image F of the multiplication map F × F → j∗A is a coherent subsheaf. On the generic fiber, we have F |X ⊆ F |X . This implies Q := (F + F )/F is a coherent sheaf on X such that Q|X =0. Upper semicontinuity shows that Q vanishes on an open neighbourhood X of the generic fiber. F |X ∩X is a locally free sheaf of OX ∩X -algebras. By Corollary 2.5, the non-Azumaya locus is again a closed subset.

Sec. 4] computing the brauer–manin obstruction 129

4. Computing the Brauer–Manin obstruction—the general strategy

4.1. In this section, we want to illustrate that the Manin pairing ev(α, x) is eﬀectively computable in certain cases. We will assume that X is a geometrically integral scheme that is proper over a number ﬁeld K. Further, let us suppose that Br (XK )=0and Br(X)=Br (X). Under these assumptions, ∗ ∼ 1 Br(X)/π Br(Spec K) = H Gal(K/K), Pic(XK ) .

These assumptions are fulﬁlled, e.g., for X a smooth proper surface such that XK is a (not necessarily minimal) rational surface. ∅ We may also suppose X( K) = as, otherwise, the Brauer–Manin obstruction would not be of much interest.

4.2. Usually, α ∈ Br(X) is not given itself, but only its image ∈ ∗ ∼ 1 α Br(X)/π Br(Spec K) = H Gal(K/K), Pic(XK ) . In order to compute ev(α, x)= ν evν (α, xν ), it is therefore necessary to explicitly lift α to Br(X). Note that the evν , as opposed to ev itself, do not factor via Br(X)/π∗Br(Spec K).

4.3. Lemma. Let K be a number ﬁeld, and let π : X → Spec K be a geometrically integral scheme that is proper over K. Then there is a natural isomorphism

∼ →i ∗ −→= 1 ker(Br (X) Br (XK ))/π Br(Spec K) H Gal(K/K), Pic(XK ) induced by the commutative diagram

Br(Spec K) Br(Spec K)

∗ π / →i / 2 ∗ / 2 0 ker(Br (X) Br (XK )) H Gal(K/K),Q(XK ) H Gal(K/K), Div(XK )

 / 1 / 2 ∗ ∗ / 2 0 H Gal(K/K), Pic(XK ) H Gal(K/K),Q(XK ) /K H Gal(K/K), Div(XK ) .

 0 Here, the bottom row is part of the long exact sequence of cohomology associated to the exact sequence

−→ ∗ ∗ −→ −→ −→ 0 Q(XK ) /K Div(XK ) Pic(XK ) 0 .

130 an application: the brauer–manin obstruction [Chap. IV

The middle column is part of the long exact cohomology sequence associated to the exact sequence

−→ ∗ −→ ∗ −→ ∗ ∗ −→ 0 K Q(XK ) Q(XK ) /K 0 .

Finally, the middle row is obtained when mapping the short exact sequence from Proposition III.5.8 to −→ 2 ∗ −→ 2 ∗ −→ 0 H Gal(Q(XK )/K),Q(XK) H Gal(Q(XK )/K),Q(XK) 0 and taking kernels.

4.4. Remark. The isomorphism given in Lemma 4.3 is the same as the isomorphism provided by the Hochschild–Serre spectral sequence (Proposi- tion III.8.11). This observation is due to S. Lichtenbaum [Lic, Sec. 2].

4.5. The diagram above leads to the following general strategy. General strategy for computing ev(α, x). a) Compute the image of α in H2 Gal(K/K),Q(X )∗/K∗ . K 2 ∗ b) Lift that to a cohomology class in H Gal(K/K),Q(XK) . c) For each ν, i) restrict to H2 Gal(K /K ),Q(X )∗ ; ν ν Kν ii) in a neighbourhood U of x such that Pic(U )=0, use the exact sequence ν Kν 2 O∗ −→ 2 ∗ H Gal(Kν /Kν ), Γ(UK , U ) H Gal(Kν /Kν ),Q(XK ) ν Kν ν −→ H2 Gal(K /K), Div(U ) ν Kν 2 O∗ in order to lift to H Gal(Kν /Kν ), Γ(UK , U ) ; ν Kν iii) apply the evaluation map Γ(U , O∗ ) → K∗ at x to get a cohomology class Kν U ν ν 2 ∗ Kν

in H Gal(Kν /Kν ), Kν ; iv) take its invariant in É/ . d) Take the sum of all these invariants.

4.6. Remark. In practice, there are several problems with that strategy. First of all, to describe a general Galois cohomology class seems to require an infinite amount of data. However, for some suitable finite field extension L/K, ∈ 1 the class α H Gal(K/K ), Pic(XK ) is the image under inflation of an element 1 in H Gal(L/K), Pic(XL) .

4.7. Observations (Inﬂation from a ﬁnite quotient Gal(L/K)). a) There is an exact sequence

∗ ∗ 0 −→ Q(XL) /L −→ Div(XL) −→ Pic(XL) −→ 0 1 2 ∗ ∗ inducing a map H Gal(L/K), Pic(XL) → H Gal(L/K),Q(XL) /L .

Sec. 4] computing the brauer–manin obstruction 131 2 ∗ 2 ∗ ∗ b) The homomorphism H Gal(L/K),Q(XL) → H Gal(L/K),Q(XL) /L is not surjective, in general. It is, however, when Gal(L/K) is cyclic. Indeed, in this case there is a non-canonical isomorphism 3 ∗ ∼ 1 ∗ H Gal(L/K),L = H Gal(L/K),L and the latter group vanishes by Hilbert’s Theorem 90. 2 ∗ c) The restriction in i) goes to H Gal(Lw/Kν ),Q(XLw ) for w a prime above ν. In step ii), one can work with U = Spec O . There is the exact sequence XLw ,xν 2 O∗ −→ 2 ∗ H Gal(Lw/Kν ), Γ(ULw , U ) H Gal(Lw/Kν ),Q(XLw ) Lw −→ 2 H Gal(Lw/K), Div(ULw ) 2 ∗ such that one may lift to H Gal(Lw/Kν ), Γ(UL , O ) . w ULw

4.8. Remark. To summarize, we assume that Gal(L/K) is cyclic and start 1 with a class in the image under inﬂation of H Gal(L/K), Pic(XL) .Thenallthe cohomology classes obtained according to the general strategy are in the image 2 2 of H Gal(L/K), · or H Gal(Lw/Kν ), · , respectively.

4.9. Thus, let us assume that G := Gal(L/K) is a cyclic group of order n.

2 ∼

In this case, H (G, ) = /n and the cup product with a generator induces an isomorphism Hˆ q(G, A) → Hˆ q+2(G, A) for all integers q and all G-modules. We ﬁx 2 a generator of H (G, ) in order to determine these isomorphisms uniquely. 2

Note that there is no distinguished generator of H (G, ) unless a generator of G is ﬁxed. Thus, the isomorphisms discussed above are not canonical. Nevertheless, there is a commutative diagram, analogous to the one in Lemma 4.3, with H2 replaced by Hˆ 0 and H1 replaced by Hˆ −1.

4.10. We therefore have the following plan. Plan for computing ev(α, x). One has

−1 ∼ 0 G 0 Hˆ (G, Pic(XL)) = [Div (XL) ∩ NDiv(XL)]/N Div (XL) .

0 G 0 The map from step a) becomes the canonical map to Div (XL) /N Div (XL). G Step b) is the lift to Q(XL) /N Q(XL)=Q(XK )/N Q(XL) under f → div f. For each ν, steps c.i), ii) and iii) amount to the evaluation

−→ ∗ ∗ Q(XK )/N Q(XL) Kν /N Lw ,

f → [f(xν )] at xν . Note that this map is well deﬁned although a representative in Q(XK ) might have a pole at xν .

132 an application: the brauer–manin obstruction [Chap. IV

Finally, in step c.iv), one has to apply the chosen isomorphism backwards,

∼

∗ ∗ ˆ 0 ∗ = 2 ∗ invν

−→ −→ É Kν /N Lw = H (G, Lw) H (G, Lw) / . ev(α, x) is the sum of all these local invariants.

4.11. Remark. Our assumption that Br (XK )=0is clearly a limitation. However, in cases when Br (XK ) is non-zero, one might want to restrict considerations to the Brauer classes in the kernel of the canonical homomorphism → Br (X) Br (XK ) and otherwise work as above. This is what is called the algebraic Brauer–Manin obstruction. On the other hand, non-algebraic Brauer classes also have the potential to ob- struct against weak approximation or the Hasse principle. This was discovered by D. Harari [Har94] and is called the transcendental Brauer–Manin obstruction. The transcendental Brauer–Manin obstruction is technically more diﬃcult and less understood than the algebraic one. To get an impression of the eﬀorts that are necessary, we refer the reader to Harari’s paper, E. Ieronymou [Ie],andT.Preu[Pr].

5. The examples of Mordell i. Formulation of the results.

5.1. In this section, we present a series of examples of cubic surfaces

over É, for which the Hasse principle fails. Our series generalizes the examples

Ðº º½¹º of Mordell [Mord]. It was observed by Yu. I. Manin [Man, VI, ] himself that the failure of the Hasse principle in Mordell’s examples may be explained by the Brauer–Manin obstruction.

5.2. Let p0 ≡ 1(mod 3) be a prime number, and let K/É be the unique É cubic ﬁeld extension contained in the cyclotomic extension É(ζp0 )/ .Weﬁxthe explicit generator θ ∈ K given by

−

θ := trÉ (ζ 1) . (ζp0 )/K p0

More concretely, we write − i θ = 2n + ζp0 , ∗ 3 i∈( ) p0 where n is given by p0 =6n +1.

5.3. Proposition. Consider the cubic surface X ⊂ P3 , given by É 3 (i) (i) 2 x3(a1x0 + d1x3)(a2x0 + d2x3)= x0 + θ x1 +(θ ) x2 . i=1

(i) É Here, a1,a2,d1,d2 ∈ .Theθ are the images of θ under Gal(K/ ).

Sec. 5] the examples of mordell 133

2 2 i) Let A ≡ 1(mod 3) be the unique integer such that 4p0 = A +27B . Suppose that d1d2 =0 , a1 =0 , a2 =0 , a1/d1 = a2/d2, and that

2 4p0−A x(a1 + d1x)(a2 + d2x)+ 2− − 2 =0 (p0 3p0 A) has no multiple zeroes. Then X is smooth. ii) Assume that p0 d1d2, that gcd(a1,d1) and gcd(a2,d2) contain only prime fac-

tors that decompose in K, and that x(a1 + d1x)(a2 + d2x) − 1=0has at least one

 ∅ simple zero in p0 .ThenX( É ) = . iii) Assume that X is smooth. Suppose further that p0 d1d2 and gcd(d1,d2)=1. Then there is a class α ∈ Br(X) with the following property.

For an adelic point x =(xν )ν ,thevalueof ev(α, x) depends only on the component x .Writex =: (t : t : t : t ). Then one has ev(α, x)=0if and νp0 νp0 0 1 2 3 only if a1t0 + d1t3 t3 ∗ is a cube in p0 .

5.4. Remarks. i) K/É is an abelian cubic field extension. It is totally ramified at p0 and unramified at all other primes. A prime p is completely decomposed in K if and only if p is a cube modulo p0. 3 (i) (i) 2 2 ii) We have i=1 x0 + θ x1 +(θ ) x2 = NK/É (x0 + θx1 + θ x2) . 2 4p0−A iii) The reduction of x(a1 + d1x)(a2 + d2x)+ 2− − 2 modulo p0 is exactly (p0 3p0 A)

x(a1 + d1x)(a2 + d2x) − 1. ∈ É 5.5. Remark. Let (t0 : t1 : t2 : t3) X( p0 ). Then, for the reduction, one has (t0 : t3)=(1:s) for s a solution of x(a1 + d1x)(a2 + d2x) − 1=0.

On the other hand, if s is a simple solution, then, by Hensel’s lemma, there ex- ∈ É ists (t0 : t1 : t2 : t3) X( p0 ) such that (t0 : t3)=(1:s).

The possible values of s are in bijection with the up-to-three planes a p0-adic point on X may reduce to.

We will show in Fact 5.19 that no Ép0 -valued point on X reduces to the triple line “x0 = x3 =0”.

5.6. Observation. As a t + d t a + d s 1 0 1 3 = 1 1 , t3 s the value of ev(α, x) depends only on the plane, to which its component x is νp0 mapped under reduction.

5.7. Remark. Since s is a solution of x(a1 +d1x)(a2 +d2x)−1=0,wehave

 a1+d1s s =0and a1 + d1s =0. Hence, s is well deﬁned and non-zero in p0 .

134 an application: the brauer–manin obstruction [Chap. IV ii. Smoothness.

5.8. Lemma. The minimal polynomial of θ is p − 1 p3 − 3p2 − Ap x3 + p x2 + p 0 x + 0 0 0 . 0 0 3 27

5.9. Remark. This is an Eisenstein polynomial. We see once more that 1 K/É is totally ramiﬁed at p0.Further,ν(θ)= 3 for ν the extension of νp0 to K. ∗ 5.10. Proof of the lemma. p0 is decomposed into three cosets C1, C2, ∗ 3 and C3 when factored by ( p0 ) .Wehave (i) − j θ = 2n + ζp0 . j∈Ci

As the sum over all p0-th roots of unity vanishes, this immediately implies (1) (2) (3) θ + θ + θ = −p0. Further, multiplying terms and adding-up shows that p − 1 (1) (2) (2) (3) (3) (1) 2 − j 0 j θ θ + θ θ + θ θ =12n +2( 2n) ζp0 + ζp0

∗ 3 ∗

∈ ∈ j p j p 4 5 0 0 p − 1 2 p − 1 p − 1 =12 0 +2 0 − 0 6 3 3 p − 1 = p · 0 . 0 3

− 3 2 (1) (2) (3) p0+3p0+Ap0 In order to establish θ θ θ = 27 , multiplying expressions leads to p − 1 (1) (2) (3) − 3 2 j − 0 j θ θ θ = 8n +4n ζp0 2n ζp0

∗ 3 ∗ ∈ j∈ j p0 p0 j j j + ζp0 ζp0 ζp0 j∈C1 j∈C2 j∈C3 − 3 j j j = 8n + ζp0 ζp0 ζp0 . j∈C1 j∈C2 j∈C3 To calculate the remaining triple product is more troublesome as one needs to know j1 j2 j3 ∈ how often a product ζp0 ζp0 ζp0 for ji Ci is equal to one. − p 1 · p0+1+A Sublemma 5.11 shows that this happens exactly 3 9 times. As the expres-

(p−1)3 É sion is invariant under Gal( É(ζp0 )/ ) and there are all in all 27 summands, we ﬁnd − j j j p0 1 ζ ζ ζ = (p0 +1+A) p0 p0 p0 27 j∈C1 j∈C2 j∈C3 1 − 2 − j + [(p0 1) (p0 +1+A)] ζp0 27 ∗

j∈ p0 3p − 1+Ap = 0 0 . 27

Sec. 5] the examples of mordell 135

− −p3+3p2−3p +1 − 3 − p0 1 3 0 0 0 Further, 8n = 8( 6 ) = 27 . The assertion follows.

∗ 5.11. Sublemma. Let D be a non-cube in p0 .Then { ∈ 2 | 3 2 3 } # (x, y) p0 Dx + D y =1 = p0 +1+A.

Proof. The number of solutions in p0 of such an equation may be counted using Jacobi sums. As in [I/R, Chapter 8, §3], we see

2 2 #{(x, y) | Dx3 + D2y3 =1} = χi(a/D)χj(b/D2) i=0 j=0 a+b=1 2 2 = χi(1/D)χj(1/D2) J(χi,χj ) , i=0 j=0 where χ is a ﬁxed cubic character. The summands for i = j are the same as in the case D =1.Ifi =0and j =0 or vice versa, then the corresponding summand is 0. Finally, by [I/R, Chapter 8, §3, Theorem 1.c)], we have J(χ, χ2)=J(χ2,χ)=−1. Altogether,

#{(x, y) | Dx3 + D2y3 =1} =#{(x, y) | x3 + y3 =1} +3.

The claim now follows from a theorem of C. F. Gauß [I/R, Chapter 8, §3, Theo- rem 2)].

5.12. Notation. We will write 3 (i) (i) 2 F (x0,x1,x2,x3):=x3(a1x0 + d1x3)(a2x0 + d2x3) − x0 + θ x1 +(θ ) x2 . i=1

5.13. Proof of Proposition 5.3.i) Since d1d2 =0 , the projection to the

first three coordinates induces a morphism of K-schemes q : X → P2 that is finite É of degree three. The conditions ∂F =0and ∂F =0for a singular point x depend only on q(x). ∂x1 ∂x2 Let us first analyze them. (i) (i) 2 Writing li := x0 + θ x1 +(θ ) x2, we get the system of equations

(1) (2) (3) θ l2l3 + θ l3l1 + θ l1l2 =0, (1)2 (2)2 (3)2 θ l2l3 + θ l3l1 + θ l1l2 =0, the solution of which is

(2) (3) (2) (3) (3) (1) (3) (1) (1) (2) (1) (2) (l2l3 : l3l1 : l1l2)=(θ θ (θ − θ ):θ θ (θ − θ ):θ θ (θ − θ )) .

As :(x0 : x1 : x2) → (l2l3 : l3l1 : l1l2) is a quadratic transformation, there are exactly four points in P2 that fulﬁll that condition. These are the three points of indeterminacy of , which we denote by P1,P2,andP3, and a non-trivial solution P .

136 an application: the brauer–manin obstruction [Chap. IV

P1,P2,andP3 are given by li = lj =0for a pair of indices i = j. This implies that these points are not contained in the line “x0 =0”. The ﬁber of q over any of these three points is therefore given by x3(a1 + d1x3)(a2 + d2x3)=0, which shows that q is unramiﬁed. Consequently, the points over P1,P2,andP3 are smooth points of X.

It remains to consider the ﬁber of P . For that point, we ﬁnd θ(1) θ(2) θ(3) (l : l : l )= : : . 1 2 3 θ(2) − θ(3) θ(3) − θ(1) θ(1) − θ(2)

The linear system of equations

θ(1) x + θ(1)x +(θ(1))2x = , 0 1 2 θ(2) − θ(3) θ(2) x + θ(2)x +(θ(2))2x = , 0 1 2 θ(3) − θ(1) θ(3) x + θ(3)x +(θ(3))2x = 0 1 2 θ(1) − θ(2) has the obvious solution

(x0 : x1 : x2) =(−3θ(1)θ(2)θ(3) :2[θ(1)θ(2) + θ(2)θ(3) + θ(3)θ(1)]:[−(θ(1) + θ(2) + θ(3))]) 3 − 2 − p0 3p0 Ap0 2 = : (p0 − 1)p0 : p0 9 3 6(p − 1) 9 = 1: 0 : , 2 − − 2 − − p0 3p0 A p0 3p0 A which is unique since the coeﬃcient matrix is Vandermonde.

Using Lemma 5.8 again, a direct calculation, which is conveniently done in maple, shows that 3 6(p − 1) 9 4p − A2 1+ 0 θ(i) + (θ(i))2 = − 0 . p2 − 3p − A p2 − 3p − A (p2 − 3p − A)2 i=1 0 0 0 0 0 0 Therefore, the ﬁber of q over P is given by 4p − A2 x (a + d x )(a + d x )+ 0 =0. 3 1 1 3 2 2 3 2 − − 2 (p0 3p0 A) By assumption, this equation has no multiple solutions. Therefore, q is unramiﬁed over P . The points above P are hence smooth points of X.

5.14. Remark. The fact that X is smooth is not mentioned in the original literature, neither in [Mord] nor in [Man].

Sec. 5] the examples of mordell 137 iii. Existence of an adelic point.

 ∅ 5.15. Proof of Proposition 5.3.ii) We have to show that X( Éν ) = for

every valuation of É.

 ∅ É ∅ X( Ê) = is obvious. For a prime number p,inordertoshowX( p) = ,we use Hensel’s lemma. It is suﬃcient to verify that the reduction Xp has a smooth

p-valued point.

Case 1: p = p0.

As νp0 (θ) > 0, the reduction Xp0 is given by

3 x3(a1x0 + d1x3)(a2x0 + d2x3)=x0.

This is the union of three planes meeting in the line given by x0 = x3 =0.Byas- 2 sumption, one of them has multiplicity one and is deﬁned over p0 . It contains p0 smooth points.

Case 2: p = p0,p d1d2. It suﬃces to show that there is a smooth p-valued point on the intersection Xp of Xp with the hyperplane “x0 =0”. This curve is given by the equation 3 3 (1) (2) (3) (i) d1d2x3 = θ θ θ (x1 + θ x2) . i=1 If p =3, then this equation deﬁnes a smooth genus one curve. It has an p-valued point by Hasse’s bound. → 1 → If p =3, then the projection Xp P given by (x1 : x2 : x3) (x1 : x2) is 3 (i) one-to-one on p-valued points. At least one of them is smooth since i=1(x+θ ) is a separable polynomial. Case 3: p = p ,p|d d . 0 1 2 ∩ (1) (2) (3) 3 (i) Xp := Xp “x0 =0”isgivenby0=θ θ θ i=1(x1 + θ x2).Inparticular, ∈ x =(0:0:0:1) Xp( p). We may assume that x is singular.

Then Xp is given as Q(x0,x1,x2)x3 + K(x0,x1,x2)=0for Q a quadratic form ≡ and K a cubic form. If Q 0, then there is an p-rational line through x such | that Q =0 . Hence, meets Xp twice in x andonceinanotherp-valued point that is smooth.

Otherwise, (F mod p) does not depend on x3, i.e., the left-hand side of the equation of X vanishes modulo p. This means, one of the factors on the left-hand side vanishes modulo p. Say, we have a1 ≡ d1 ≡ 0(mod p). Then, by assumption, p decomposes completely in K. At such a prime, Xp is the

union of three lines that are all deﬁned over p, diﬀerent from each other, and meeting in one point. We have plenty of smooth points. iv. Construction of a Brauer class.

5.16. We write G := Gal(K/É). According to the general strategy described in the section above, an element of

138 an application: the brauer–manin obstruction [Chap. IV

1 1 É

H (G, Pic(XK )) ⊆ H Gal( É/ ), Pic(X ) may be given by a rational function É f ∈ Q(X) such that div(f) ∈ NDiv(XK ). Such a function is a x + d x f := 1 0 1 3 . x3

Indeed, “a1x0 + d1x3 =0” deﬁnes a triangle, the three lines of which are given (i) (i) 2 by that equation and x0 + θ x1 +(θ ) x2 =0for i =1, 2,or3, respectively. Considered as a Weil divisor, this triangle is the norm of the divisor given by one of the lines. We write [f] for the Brauer class deﬁned by f.

a2x0+d2x3 5.17. Remarks. i) One might work as well with [f ] for f := x or

Ðº ÈÖÐÓ Ò º even with both [f] and [f ] as Manin does in [Man, VI, ]. However, 4 5 4 5 2 · (a1x0 + d1x3)(a2x0 + d2x3) x0 + θx1 + θ x2 [f] [f ]= 2 = N =0. x3 x3

[f] [f ]

É Consequently, X( É ) = X( ) .

∗ ∼

ii) Under very mild hypotheses on X, one has Br(X)/π Br(Spec É) = /3 and [f] is a generator. We will discuss this point in Subsection vi.

5.18. Lemma. Let ν be any valuation of É diﬀerent from νp0 .Then

evν ([f],x)=0 ∈

for every adelic point x X( É ). Proof. First step. Elementary cases. 1 If ν = ν∞,thenevν ([f],x)=0since #G =3while only the values 2 and 0 are possible. ∗ If ν = νp and p is decomposed in K, then every element of Ép is a norm. There- fore, evν ([f],x)=0. Second step. Preparations. It remains to consider the case in which p remains prime in K. Wehavetoshowthat

a1t0+d1t3 ∗ ∈ É is a norm for each point (t0 : t1 : t2 : t3) ∈ X( Ép).Anelementw is t3 p a norm if and only if 3|νp(w).

It might happen that θ is not a unit in Kν .AsKν /Ép is unramiﬁed, there exists ∈ ∗ ∈ t Ép such that θ := tθ Kν is a unit. The surface X given by 3 (i) (i) 2 x3(a1x0 + d1x3)(a2x0 + d2x3)= x0 + θ x1 +(θ ) x2 i=1 → t1 t2 is isomorphic to X.Themapι:(t0 : t1 : t2 : t3) (t0 : t : t2 : t3) is an isomorphism X → X that leaves the rational function a1x0+d1x3 unchanged. Hence, we x3 may assume without restriction that θ ∈ Kν is a unit.

Sec. 5] the examples of mordell 139

Third step. The case θ is a unit.

We may assume t ,t ,t ,t ∈ are coprime. 0 1 2 3 p 3 (i) (i) 2 We could have i=1 t0 + θ t1 +(θ ) t2 =0only when t0 = t1 = t2 =0since

θ generates a cubic ﬁeld extension of Ép.Thent3(a1t0 + d1t3)(a2t0 + d2t3)=0 implies t3 =0, which is a contradiction. Therefore, both sides of the equation are diﬀerent from zero. In particular, we automatically have t =0 . 3 a1t0+d1t3 If ν t3(a1t0+d1t3)(a2t0+d2t3) =0,then is clearly a norm. Otherwise, we t3 have 3 (i) (i) 2 ν (t0 + θ t1 +(θ ) t2) > 0 . i=1

This implies that ν(t0), ν(t1), ν(t2) > 0.Thent3 must be a unit.

From the equation of X, we deduce ν(d1d2) > 0.Ifν(d2) > 0, then, according to the assumption, d1 is a unit. This shows ν(a1t0 + d1t3)=0,fromwhichthe assertion follows.

Thus, assume ν(d1) > 0.Then d2 is a unit and, therefore, ν(a2t0 + d2t3)=0. | 3 (i) (i) 2 Further, we note that 3 ν i=1(t0 + θ t1 +(θ ) t2) since the right-hand side is a norm. By consequence, 3|ν t3(a1t0 + d1t3)(a2t0 + d2t3) . a1t0+d1t3 Altogether, we see that 3|ν(a1t0 +d1t3) and 3|ν . The claim follows. t3

5.19. Fact. i) The reduction Xp0 of X at p0 is given by

3 x3(a1x0 + d1x3)(a2x0 + d2x3)=x0 .

ii) Over the algebraic closure, Xp0 is the union of three planes meeting in a triple line “x0 = x3 =0”.

iii) No Ép0 -valued point on X reduces to the triple line.

Proof. i) and ii) are clear.

∈ É ∈ iii) Let (t0 : t1 : t2 : t3) X( p0 ). We may assume t0,t1,t2,t3 p0 are coprime. ≥ ≥ We write ν for the extension of νp0 to K.Thenν(t0) 1 and ν(t3) 1 together imply ν(t3(a1t0 + d1t3)(a2t0 + d2t3)) ≥ 3. On the other hand, 3 (i) (i) 2 ν (t0 + θ t1 +(θ ) t2) i=1 (i) 1

is equal to 1 or 2,sincet1 or t2 is a unit and ν(θ )= 3 . ∈ É 5.20. Lemma. Let (t0 : t1 : t2 : t3) X( p0 ).

a1t0+d1t3 ∗ Then evνp ([f], (t0 : t1 : t2 : t3)) = 0 if and only if is a cube in . 0 t3 p0 3 Proof. (p0)=p is totally ramiﬁed. In Ép0 , there is a uniformizer being a norm. ∗ Further, a p0-adic unit u is a norm if and only if u := (u mod p0) is a cube in p0 .

140 an application: the brauer–manin obstruction [Chap. IV

a1t0+d1t3 We have that is automatically a p0-adic unit. Indeed, suppose that

t3 ∈ t0,t1,t2,t3 p0 are coprime. Modulo p0, the equation of X is

3 x3(a1x0 + d1x3)(a2x0 + d2x3)=x0 .

As no point may reduce to the singular line, we see that t0 =0 . This implies t3 =0 and a1t0 + d1t3 =0 , which is the assertion.

v. Examples.

5.21. Corollary. Let p0 ≡ 1(mod 3) be a prime number, and consider the

cubic surface X ⊂ P3 , given by É 3 (i) (i) 2 x3(a1x0 + d1x3)(a2x0 + d2x3)= x0 + θ x1 +(θ ) x2 . i=1 Here, a1,a2,d1,d2 ∈ , and assume that a1 =0, a2 =0, a1/d1 = a2/d2, p0 d1d2, and that gcd(a1,d1) and gcd(a2,d2) contain only prime factors that decompose in K. i) Assume that

x(a1 + d1x)(a2 + d2x) − 1=0

(1) (2) (3) ∈ has three diﬀerent zeroes t ,t ,t p0 . (1) (2) (3)

a1+d1t a1+d1t a1+d1t ∗

É ∅ If , ,and are non-cubes in ,thenX( )= .OnX, t(1) t(2) t(3) p0 there is a Brauer–Manin obstruction to the Hasse principle. ∗ If exactly one of the three expressions is a cube in p0 , then, on X,thereisa Brauer–Manin obstruction to weak approximation. ii) Assume that

x(a1 + d1x)(a2 + d2x) − 1=0 ∈ has exactly one zero t p0 ,whichissimple.

a1+d1t ∗

É ∅ If is not a cube in ,thenX( )= .OnX, there is a Brauer–Manin t p0 obstruction to the Hasse principle.

5.22. Remark. In the case of three diﬀerent solutions, it is impossible that (1) (2) (3) a1+d1t a1+d1t a1+d1t exactly two of the expressions t(1) , t(2) ,and t(3) are cubes. Indeed, a direct calculation shows a + d t(1) a + d t(2) a + d t(3) 1 1 · 1 1 · 1 1 = d3 . t(1) t(2) t(3) 1

This means, the Brauer–Manin obstruction might allow no, one, or all three of the planes, the reduction Xp0 consists of, but not exactly, two of them.

Sec. 5] the examples of mordell 141

5.23. Example. Let p0 ≡ 1(mod 3) be a prime number, and assume that ∗ | | d1 and d2 are non-cubes in p0 such that d1d2 is a cube, p0 a1 and p0 a2.Fur- ther, suppose a1 =0 , a2 =0 ,anda1/d1 = a2/d2,aswellasthatgcd(a1,d1) and gcd(a2,d2) contain only prime factors that decompose in K. Then the cubic surface X given by 3 (i) (i) 2 x3(a1x0 + d1x3)(a2x0 + d2x3)= x0 + θ x1 +(θ ) x2 i=1 is a counterexample to the Hasse principle. Indeed, one has a1t0 + d1t3 ≡ d1 (mod p0) , t3 which is a non-cube. The assumption that d1d2 is a cube is important to guaran- ∅ tee X( Ép0 ) = .

More concretely, for p0 =19, a counterexample to the Hasse principle is given by

3 (i) (i) 2 x3(19x0 +5x3)(19x0 +4x3)= x0 + θ x1 +(θ ) x2 i=1 3 − 2 2 2 = x0 19x0x1 + 133x0x2 + 114x0x1 − 2 − 3 1 539x0x1x2 + 5 054x0x2 209x1 2 − 2 3 + 3 971x1x2 23 826x1x2 + 43 681x2 .

5.24. Example. For p0 =19, consider the cubic surface X given by 3 (i) (i) 2 x3(x0 + x3)(12x0 + x3)= x0 + θ x1 +(θ ) x2 .

i=1

 ∅ É ∅

Then X( É ) = but X( )= .OnX, there is a Brauer–Manin obstruction to the Hasse principle.

Indeed, in 19, the cubic equation

x(1 + x)(12 + x) − 1=0

has the three solutions 12, 15,and17. However, in 19, 13/12 = 9, 16/15 = 15, and 18/17 = 10, which are three non-cubes.

5.25. Example. For p0 =19, consider the cubic surface X given by 3 (i) (i) 2 x3(x0 + x3)(6x0 + x3)= x0 + θ x1 +(θ ) x2 . i=1 Then, on X, there is a Brauer–Manin obstruction to weak approximation.

Indeed, in 19, the cubic equation

x(1 + x)(6 + x) − 1=0

has the three solutions 8, 9,and14. However, in 19, 10/9=18is a cube while 9/8=13and 15/14 = 16 are non-cubes.

142 an application: the brauer–manin obstruction [Chap. IV

− The smallest É-rational point on X is (14:15:2:( 7)). Note that indeed x3/x0 = −7/14 ≡ 9(mod 19).

5.26. Example. For p0 =19, consider the cubic surface X given by 3 (i) (i) 2 x3(x0 + x3)(2x0 + x3)= x0 + θ x1 +(θ ) x2 . i=1 Then, on X, there is a Brauer–Manin obstruction to the Hasse principle.

Indeed, in 19, the cubic equation

x(1 + x)(2 + x) − 1=0

has x =5as its only solution. The other two solutions are conjugate to each other in 192 . However, in 19, 6/5=5is a non-cube.

5.27. Example (Swinnerton-Dyer [SD62]). For p0 =7, consider the cubic surface X given by

3 (i) (i) 2 x3(x0 + x3)(x0 +2x3)= x0 + θ x1 +(θ ) x2 i=1 3 − 2 2 2 − = x0 7x0x1 +21x0x2 +14x0x1 77x0x1x2 2 − 3 2 − 2 3 +98x0x2 7x1 +49x1x2 98x1x2 +49x2 .

Then, on X, there is a Brauer–Manin obstruction to the Hasse principle.

Indeed, in 7, the cubic equation

x(1 + x)(1 + 2x) − 1=0

has x =5as its only solution. However, in 7, 6/5=4is not a cube.

5.28. Remark. From each of the examples given, by adding multiplies of p0 to the coefficients a1, d1, a2,andd2, a family of surfaces arises that are of similar nature. In Example 5.23, some care has to be taken to keep a1 and a2 different from zero and a1/d1 different from a2/d2. In all examples, gcd(a1,d1) and gcd(a2,d2) need some consideration.

5.29. Remark (Lattice basis reduction). The norm form in the p0 =19ex- amples produces coefficients that are rather large. An equivalent form with smaller coefficients may be obtained using lattice basis reduction. In its simplest form, this means the following. 3 (1) (2) (3) For the rank-two lattice in Ê , generated by the vectors v1 := (θ ,θ ,θ ) and (1) 2 (2) 2 (3) 2 v2 := (θ ) , (θ ) , (θ ) , in fact {v1,v2 +7v1} is a reduced basis. There- − fore, the substitution x1 := x1 7x2 simplifies the norm form. Actually, we find 3 (i) (i) 2 3 − 2 2 − 2 x0 + θ x1 +(θ ) x2 = x0 19x0x1 +114x0(x1) +57x0x1x2 133x0x2 i=1 − 3 − 2 2 − 3 209(x1) 418(x1) x2 + 1045x1x2 209x2 .

Sec. 5] the examples of mordell 143

∗ vi. Calculation of the complete Br(X)/π Br(Spec É).

5.30. Notation. Let X be a cubic surface in P3 given by

l1l2l3 = λ1λ2λ3 for six linear forms l1,l2,l3,λ1,λ2,λ3.

Then we write Lij for the line on X given by li = λj =0. The subgroup of Pic(X) generated by these nine lines will be denoted by P .Fur- ther, we let S be the free abelian group over all lines Lij.

5.31. Facts. i) P ⊂ Pic(X) is a lattice of rank ﬁve and discriminant 3. ii) The kernel S0 of the canonical homomorphism S → P is generated by D1 := L11 + L12 + L13 − L21 − L22 − L23, D2 := L11 + L12 + L13 − L31 − L32 − L33, D3 := L11 + L13 − L22 − L32,andD4 := L11 + L12 − L23 − L33.

Proof. We have D1 = div(l1/l2), D2 = div(l1/l3), D3 = div(l1/λ2),and D4 = div(l1/λ3).Thus,D1,D2,D3,D4 ∈ ker(S → P ). It is easy to check that D1,D2,D3,D4 are linearly independent. This shows that rk P ≤ 5. On the other hand, the intersection matrix of L , L , L , L ,andL is ⎛ 11 ⎞ 22 23 32 33 −10000 ⎜ ⎟ ⎜ 0 −1110⎟ ⎜ ⎟ ⎜ 01−101⎟ . ⎝ 010−11⎠ 0011−1 The determinant of this matrix is equal to 3, which is a square-free integer. Asser- tion i) is proven.

It remains to verify that D1, D2, D3,andD4 generate ker(S → P ). For this, we have to show that the canonical injection

L11,L22,L23,L32,L33−→S/D1,D2,D3,D4 is actually bijective.

For surjectivity, observe that, modulo D1,D2,D3,D4, the remaining generators are given by L12 ≡−L11 + L23 + L33, L13 ≡−L11 + L22 + L32, L21 ≡ L11 + L12 + L13 − L22 − L23 ≡−L11 + L32 + L33, and, ﬁnally, L31 ≡ L11 + L12 + L13 − L32 − L33 ≡−L11 + L22 + L23.

5.32. Proposition. Let K/É be a Galois extension of degree three. As- sume that λ1,λ2,λ3 are deﬁned over K and form a Galois orbit and that l1,l2,l3 are linear forms deﬁned over É.

ˆ −1

Then H Gal(K/É),P = /3 and [l1/l2] is a generator. Proof. By Lemma III.8.19, we have a canonical isomorphism ∼ ˆ −1 −→= ∩ H Gal(K/É),P (NS S0)/N S0 .

Let us calculate NS ∩ S0.

144 an application: the brauer–manin obstruction [Chap. IV

D1 = N(L11 − L21) and D2 = N(L11 − L31) are clearly norms. If

aD3 + bD4 =(a + b)L11 + bL12 + aL13 − aL22 − bL23 − aL32 − bL33 is a norm, then the coeﬃcients of L31, L32,andL33 must coincide. Hence, a = b =0. Consequently,

NS ∩ S0 = D1,D2 .

Further, N(D1)=3D1, N(D2)=3D2,andN(D3)=N(D4)=D1 + D2. Hence,

NS0 = 3D1, 3D2,D1 + D2 .

The assertion follows.

5.33. Return to the case of a cubic surface given an equation of type 3 (i) (i) 2 x3(a1x0 + d1x3)(a2x0 + d2x3)= x0 + θ x1 +(θ ) x2 . i=1

1 1 ∼

To deduce information on H Gal( É/ ), Pic(X ) from H (G, P ) = /3 ,we É need some understanding of the remaining 18 lines.

Over É(θ) there are at least nine lines deﬁned. The list of the 350 conjugacy classes of subgroups of W (E6), established using GAP (cf. III.8.23 and the Appendix), contains only four classes ﬁxing nine or more lines. The corresponding extract looks like this.

Thus, the remaining 18 lines are deﬁned over an extension of É(θ),whichmaybe of degree 6, 3, 2,or1. ≥

We see that rk Pic(X É(θ)) 5. Equality holds if and only if the ﬁeld of deﬁnition

of the 27 lines is of degree 3 or 6 over É(θ). We also observe that, in any case,

1 É

H Gal( É/ (θ)), Pic(X ) =0. É

5.34. Remark. For every concrete choice of the coefficients, one may determine the field of definition of the 27 lines by a Gröbner base calculation. It turned

out that it was of degree 6 over É(θ) in every example we tested. Thus, degree 6 seems to be the generic case.

If the ﬁeld of deﬁnition of the 27 lines is a degree 6 or degree 3 extension of É(θ),

1 É

then we may describe H Gal( É/ ), Pic(X ) completely. É

Sec. 5] the examples of mordell 145

(i) 5.35. Proposition. Let p0 ≡ 1(mod 3) be a prime number, θ as above,

and let X ⊂ P3 be the cubic surface given by É 3 (i) (i) 2 x3(a1x0 + d1x3)(a2x0 + d2x3)= x0 + θ x1 +(θ ) x2 i=1

for a1,a2,d1,d2 ∈ . Assume that the ﬁeld of deﬁnition of the 27 lines on X is of

degree 3 or 6 over É(θ).

∗ ∼ a1x0+d1x3

Then Br(X)/π Br(Spec É) = /3 and [ x ] is a generator. 3

1 É

Proof. Since H Gal( É/ (θ)), Pic(X ) =0, the inﬂation map É

1 Gal( É/ (θ) 1

É −→ É É

H Gal( É(θ)/ ), Pic(X ) ) H Gal( / ), Pic(X )

É É

is an isomorphism. Hence, in view of Proposition 5.32, it will suﬃce to show that É

P = Pic(X )Gal( É/ (θ)). É “⊆”isobvious. ⊇

“ ” Our assumption implies that rk Pic(X É(θ))=5. Further, from Proposi-

tion III.8.11, we see that Pic(X É(θ)) is always a subgroup of ﬁnite index in

É É É

Pic(X )Gal( É/ (θ)). Hence, rk Pic(X )Gal( / (θ)) =5too.

É É É

By Fact 5.31.i), we have rk P =5.Thus,P ⊆ Pic(X )Gal( É/ (θ)) is a sublattice of É ﬁnite index. Since Disc P =3is square-free, the claim follows.

5.36. Example. For p0 =19, consider the cubic surface X given by 3 (i) (i) 2 x3(x0 + x3)(7x0 + x3)= x0 + θ x1 +(θ ) x2 . i=1 Then, on X, there is no Brauer–Manin obstruction to weak approximation. Indeed, a Gröbner base calculation shows that the 27 lines on X are deﬁned over

∗ ∼

É adegree6 extension of É(θ).Thus,Br(X)/π Br(Spec ) = /3 . A generator is given by the class α := [ x0+x3 ]. x3

However, in 19, the cubic equation

x(1 + x)(7 + x) − 1=0

has the only solution x =11. Hence, every É19-valued point on X has a reduction of x0+x3 the form (1 : y : z : 11). For the local evaluation, we ﬁnd =12/11 = 8 ∈ 19, x3

which is a cube. ∈

Thus, ev(x, α)=0for every adelic point x X( É ). By consequence,

É X( É ) = X( ). One might expect that X satisﬁes weak approximation.

5.37. Remarks (Degenerate cases). i) If the ﬁeld of deﬁnition of the 27 lines

∗ É is quadratic over É(θ),thenBr(X)/π Br(Spec )=0. In fact, in this case, we have a cyclic group G of order 6 acting on the 27 lines. There are seven conjugacy classes of cyclic groups of order 6 in W (E6).Wehave

146 an application: the brauer–manin obstruction [Chap. IV the following list.

26#U=6[2,3],#H^1 = 1, Rk(Pic) = 1, Orbits [ 3, 6, 6, 6, 6 ] 27#U=6[2,3],#H^1 =4[2,2],Rk(Pic) = 1, Orbits [ 3, 6, 6, 6, 6 ] 28#U=6[2,3],#H^1 = 1, Rk(Pic) = 4, Orbits [ 1, 1, 1, 2, 2, 2, 3, 3, 3, 3, 6 ] 30#U=6[2,3],#H^1 = 1, Rk(Pic) = 3, Orbits [ 1, 1, 1, 2, 2, 2, 6, 6, 6 ] 32#U=6[2,3],#H^1 = 1, Rk(Pic) = 2, Orbits [ 3, 3, 3, 3, 3, 6, 6 ] 36#U=6[2,3],#H^1 = 1, Rk(Pic) = 3, Orbits [ 1, 2, 2, 2, 2, 3, 3, 6, 6 ] 39#U=6[2,3],#H^1 = 1, Rk(Pic) = 2, Orbits [ 3, 6, 6, 6, 6 ]

As the element of order two ﬁxes 15 lines, we may not have more than two orbits of size 6. This implies the claim. Actually, we are in class No. 32.

ii) If the 27 lines are deﬁned over É(θ), then, seemingly, there are two cases. ∗

One might either have rk Pic(X)=3.ThenBr(X)/π Br(Spec É)=0.

∗ 2

Or,otherwise,rkPic(X)=1.Inthiscase,Br(X)/π Br(Spec É)=( /3 ) . α := [ a1x0+d1x3 ] is one of the generators. x3

We do not know of an example over É, in which any of these degenerate cases occurs.

6. The “ﬁrst case” of diagonal cubic surfaces

6.1. To conclude this chapter, we report about another case, in which the eﬀect of the Brauer–Manin obstruction has been studied. Consider smooth diagonal

cubic surfaces in P3 , given by an equation of the form É

3 3 3 3 a0x0 + a1x1 + a2x2 + a3x3 =0 \{ } for a0,...,a3 ∈ 0 , such that the following additional condition is satisﬁed. ∗ | 3| | ( ) There exists a prime number p0 such that p0 a3 but neither p0 a3 nor p0 ai for i =0, 1, 2.

6.2. X has a model over Spec given by the same equation. We will denote that model by X .

On X, there is the smooth genus one curve E given by the equation x3 =0. ∗ If p0 =3, then assumption ( ) implies that the reduction Xp0 is a cone over Ep0 .

There is the mapping −→ red: X( Ép0 ) Ep0 ( p0 )

(x0 : x1 : x2 : x3) → ((x0 mod p0):(x1 mod p0):(x2 mod p0)) .

6.3. Remark. The Brauer–Manin obstruction on diagonal cubic surfaces was studied by J.-L. Colliot-Thélène, D. Kanevsky, and J.-J. Sansuc in [CT/K/S].

∗ Br

For the case when ( ) is fulﬁlled, the main result asserts that X( É ) is exactly

one third of the whole of X( É ) in a sense made precise. More concretely, there is the following theorem.

Sec. 6] the “first case” of diagonal cubic surfaces 147

6.4. Theorem (Colliot-Thélène, Kanevsky, and Sansuc). Let X ⊂ P3 be a

 ∅ ∗

smooth diagonal cubic surface such that X( É ) = . Suppose that ( ) is fulﬁlled for a certain prime number p0.

∗ ∼

a) Then Br(X)/π Br(Spec É) = /3 . b) The image of the evaluation map

× → É

ev: Br(X) X( É ) /

1 is 3 / .

c) Choose an adelic point (xν )ν ∈ X( ). ∈ ∗ Let α Br(X) be such that its image in Br(X)/π Br(Spec É) is non-trivial. ∗

Assume that α is a normalized modulo π Br(Spec É) in such a way that ev (α, x )=0. νp0 νp0 i) Assume p0 =3 .

Then the local evaluation map

→ É ev : Br(X) × X( É ) / νp0 p0 has the following property: ev (α, ·) is the composition of red with a surjective group homomorphism νp0

→ (E ( ), x ) / . p0 p0 νp0 3

In particular, ev (α, x) depends only on the reduction x ∈ X( ). νp0 p0 Further, we have

{ ∈ X | } { ∈ X | } # y ( ) ev (α, y)=0 =#y ( ) ev (α, y)=1/3 p0 νp0 p0 νp0 { ∈ X | } =#y ( ) ev (α, y)=2/3 . p0 νp0 ii) Let p0 =3.

Then the local evaluation map

× É → É evν3 : Br(X) X( p0 ) / has the following properties:

· É → É The image of evν3 (α, ): X( 3) / is 3 / .

There exists a positive integer n such that evν3 (α, x) depends only on the reduction

(3n) n x ∈ X ( /3 ).

For every y0 ∈ X ( 3), we have

{ ∈ X n | } # y ( /3 ) y = y0, evν3 (α, y)=0

{ ∈ X n | } =# y ( /3 ) y = y0, evν3 (α, y)=1/3

{ ∈ X n | } =# y ( /3 ) y = y0, evν3 (α, y)=2/3 .

148 an application: the brauer–manin obstruction [Chap. IV

In particular,

n n

| } { ∈ X | } { ∈ X # y ( /3 ) evν3 (α, y)=0 =#y ( /3 ) evν3 (α, y)=1/3

{ ∈ X n | } =#y ( /3 ) evν3 (α, y)=2/3 .

6.5. Corollary. Let X ⊂ P3 beasmoothdiagonalcubicsurfacesuch

∅ ∗ that X( É ) = . Suppose that ( ) is fulﬁlled for a certain prime number p0. a) Then, on X, there is a Brauer–Manin obstruction to weak approximation. b) There is, however, no Brauer–Manin obstruction to the Hasse principle.

6.6. Remark. Colliot-Thélène, Kanevsky, and Sansuc verify the same behaviour in their “ﬁrst case”, which is a bit more general than assumption (∗). An- other particular case, in which the same result is true, is when p0 a0,a1, but | 2 p0 a2,a3, p0 a2,a3 and a2/a3 is a p0-adic cube. The three authors also show that, in the “Second case” there exist diagonal cubic

 ∅ ∅

É É surfaces such that X( É ) = and X( ) = X( ) or . From Theorem 6.4, it is clear that this may happen only under rather restrictive conditions. For example, every prime number dividing one of the coeﬃcients must necessarily divide a second one.

6.7. Remark. The statement that the three values are equally distributed was not formulated in [CT/K/S].

∗ ∼

6.8. Remark. In order to prove Br(X)/π Br(Spec É) = /3 , we make ∗ heavy use of the list of all possible quotients Br(X)/π Br(Spec É) established with the help of a computer. Cf. III.8.23 and, for the list in full, the Appendix.

∗ i. Calculating Br(X)/π Br(Spec É).

6.9. Notation. Following Manin [Man], we will denote the lines on X as follows. We ﬁx third roots of a0,...,a3, once and for all.

Then Lk(m, n) is the line given by

1 1 3 m 3 ai xi + ζ3 aj xj =0, 1 1 3 n 3

ak xk + ζ3 a3 x3 =0. Here, k ∈{0, 1, 2}, i, j ∈{0, 1, 2}\{k}, i

6.10. In this notation, the 45 triangles on X may easily be described. 1 1

3 n 3

∈{ } ∈ i) The planes “ak xk + ζ3 a3 x3 =0”, for k 0, 1, 2 , n /3 , deﬁne nine 1 triangles Δkn. They consist of Lk(0,n), Lk(1,n),andLk(2,n). 1 1 3 m 3 ∈{ } ii) Analogously, the planes “ai xi + ζ3 aj xj =0”, for i, j 0, 1, 2 , i

∈ 2 and m /3 , deﬁne nine triangles Δkm. They consist of Lk(m, 0), Lk(m, 1), and Lk(m, 2) for k ∈{0, 1, 2}\{i, j}.

Sec. 6] the “first case” of diagonal cubic surfaces 149

1 1 1 1 3 a 3 b 3 c 3 iii) Finally, the planes given by a0 x0 + ζ3 a1 x1 + ζ3a2 x2 + ζ3a3 x3 =0for ∈{ } 3 − − a, b, c 0, 1, 2 deﬁne 27 triangles Δabc. Those consist of L0(b a, c), L1(b, c a), and L2(a, c − b). 3 3 3 6.11. Notation. We write L := É( a3/a0, a2/a0, a1/a0,ζ3) and

3 3 É K := É( a2/a0, a1/a0,ζ3).ThenG := Gal(L/ ) is the Galois group acting on

the 27 lines. We have the subgroup G1 := Gal(L/K), which is isomorphic to /3 by consequence of assumption (∗). We choose a generator σ of G1. 2 Finally, we write τ for the involution ζ3 → ζ ﬁxing the three third roots. ∼ 3 Then σ, τ = S3.

6.12. Proposition. The restriction

H1(G, Pic(X )) −→ H1(σ, Pic(X ))

É É is not the zero map.

∗ 3 É Proof. The assumption ( ) guarantees that [É( a3/a0,ζ3): ]=6. The adjunc- 3 3 tions of a1/a0 and a2/a0 might or might not lead to further cubic extensions. Thus, there will be three cases to distinguish, #G may be either 6 or 18 or 54. We take for S the free group generated by the 27 lines and use the

1 ∼

canonical isomorphisms H (G, Pic(X )) = Hom((NGS ∩ S0)/NGS0, É/ ) and É

1 ∼ H (σ, Pic(X )) = Hom((N S ∩ S0)/N S0, É/ ). The restriction is then in- É σ σ → duced by the norm map NG/σ : NσS NGS. 1 1 We will start with the divisor D := div (x0 +(a1/a0) 3 x1)/(x0 +(a2/a0) 3 x2) that is the norm of L2(0, 0) − L1(0, 0).WehavetoshowthatNG/σD is not the norm NG of a principal divisor. First case. #G =54. This is the generic case. We ﬁnd that 3 3 3 3 N D =6div (a x + a x )/(a x + a x ) G/ σ 0 0 1 1 0 0 2 2 =6 L2(m, n) − 6 L1(m, n) .

2 2

∈ (m,n)∈(/3 ) (m,n) ( /3 )

On the other hand, the group S0 of all principal divisors in S is generated by the

Ðº ÄÑÑ º pairwise diﬀerences of all triangles [Man, VI, ]. We have 1 NGΔkn =18 Lk(i, j) ,

∈ (i,j) ( /3 ) 2 NGΔkm =18 Lk(i, j) ,

∈ (i,j) ( /3 ) 3 NGΔabc =6 L0(i, j)+6 L1(i, j)+6 L2(i, j),

2 2 2

∈ ∈ (i,j)∈( /3 ) (i,j) ( /3 ) (i,j) ( /3 ) and we see that NG/σD is not generated by these elements.

150 an application: the brauer–manin obstruction [Chap. IV

Second case. #G =18. ∗

Assume without restriction that a0 = a1 and a2/a0 is not a cube in É . Then, we ﬁnd 3 3 6 3 3 3 2 N D = div (x + x ) /(x +(a /a ) x ) G/ σ 0 1 0 2 0 2 =6[L2(0, 0) + L2(0, 1) + L2(0, 2)] − 2 L1(i, j) .

2 (i,j)∈( /3 ) On the other hand, ⎧ ⎪ ⎨⎪ 2 Lk(i, j) if k =2 ,

2 (i,j)∈( /3 ) NGLk(m, n)= ⎪

⎩⎪ 6 Lk(m, j) if k =2. j∈/3

Hence, 1 NGΔkn =6 Lk(i, j),

∈ (i,j⎧) ( /3 ) ⎪ ⎨⎪ 6 Lk(i, j) if k =2 ,

2 2 (i,j)∈( /3 ) NGΔ = km ⎪

⎩⎪ 18 Lk(m, j) if k =2,

∈ j /3 3 NGΔabc =2 L0(i, j)+2 L1(i, j)+6 L2(a, j) .

2 2

∈ ∈ (i,j)∈( /3 ) (i,j) ( /3 ) j /3

Factoring modulo all summands of type L2 and ignoring the round brackets, these expressions are 54L0, 54L1,and18L0 +18L1. They do not generate (−18)L1. Third case. #G =6. Here, we ﬁnd that

1 1 N D =2div (x +(a /a ) 3 x )/(x +(a /a ) 3 x ) G/ σ 0 1 0 1 0 2 0 2

=2 L2(0,i) − 2 L1(0,i).

∈ i∈/3 i /3

On the other hand, 1

NGΔkn =2 Lk(0,i)+2 Lk(1,i)+2 Lk(2,i)

∈ ∈ ∈ i /3 i /3 i /3 2

NGΔkm =6 Lk(m, i)

∈ i /3 3 −

NGΔabc =2 L0(b a, i)+2 L1(b, i)+2 L2(a, i) .

∈ ∈ i∈/3 i /3 i /3

Ignoring the round brackets, those are 18Lk for k ∈{0, 1, 2} and 6L0 +6L1 +6L2, which do not generate 6L2 − 6L1.

Sec. 6] the “first case” of diagonal cubic surfaces 151

6.13. Corollary. The restriction map

H1(G, Pic(X )) −→ H1(σ, τ , Pic(X ))

É É is an isomorphism. The groups are isomorphic to /3 .

Proof. H1(σ, Pic(X )) is purely 3-torsion. Sir Peter Swinnerton-Dyer’s list É

(cf. III.8.24) shows it is either /3 or ( /3 ) . Further, we know that the re-

striction H1(G, Pic(X )) → H1(σ, Pic(X )) is not the zero map. Thus, the list

É É

1 2

implies that H (G, Pic(X )), too, is nothing but /3 or ( /3 ) . É

The restriction H1(G, Pic(X )) → H1(σ, Pic(X )) factors via

É É

H1(G, Pic(X )) → H1(σ, τ , Pic(X )).

É É

Thus, the latter is not the zero map, either. Again, a view on the list makes sure

1 2

that H (σ, τ , Pic(X )) may be only /3 or ( /3 ) . É

2 At this point, recall that ( /3 ) occurs only when the group acting non-trivially

on Pic(X ) is of order three. Since τ ∈σ, τ ⊂G is of order two and acting É non-trivially, we are not in this case. The assertion follows.

6.14. Proposition. The restriction map

H1(G, Pic(X )) −→ H1(σ, Pic(X ))τ

É É is an isomorphism. The groups are isomorphic to /3 .

Proof. It suﬃces to show that the restriction map

H1(σ, τ , Pic(X )) −→ H1(σ, Pic(X ))τ

É É is an isomorphism. We know, the group on the left-hand side is isomorphic to /3 . The group on the right-hand side is clearly a 3-torsion group. We consider the inﬂation-restriction spectral sequence

Ep,q := Hp(τ,Hq(σ, Pic(X ))) =⇒ Hn(σ, τ , Pic(X )) É 2 É and obtain the following exact sequence of terms of lower order

0 −→ H1(τ, Pic(X )σ) −→ H1(σ, τ , Pic(X ))

É É

−→ H1(σ, Pic(X ))τ −→ H2(τ, Pic(X )σ) .

É É

Since the homomorphism considered is encircled by 2-torsion groups, the proof is complete.

152 an application: the brauer–manin obstruction [Chap. IV ii. Some observations.

6.15. Lemma. The image of the Manin map

× → É

ev: Br(X) X( É ) /

1 is contained in 3 / . Proof. By Proposition 2.3.b.i) and iii), the Manin map is additive in the ﬁrst

∗ ∼

variable and factors via Br(X)/π Br(K) = /3 .

∈ ⊂

6.16. Lemma. Let α Br(X) be any Brauer class, and let (C, x ) X É 0 p0 beasmoothellipticcurve. Then one of the following two statements is true.

∗ · É i) α| ∈ Br(C)/π Br(Spec É) is zero. Then ev (α, ) is constant on C( ). C νp0 p0 ∗ ii) α|C ∈ Br(C)/π Br(Spec É) is non-zero.

→ Then there is a surjective group homomorphism g : C( Ép0 ) 3 / such that

ev (α, x)=g(x)+ev (α, x ) νp0 νp0 0 ∈ for every x C( Ép0 ). Proof. i) is clear. ii) follows directly from Theorem III.8.16.

6.17. Lemma. Let p0 =3 be a prime number, let (C, O) be an elliptic curve C over Ép0 ,andlet be a minimal model of C over p0 . Assume that there are no

Ép0 -valued points on C with singular reduction. Then every continuous group homomorphism

−→ g : C( É ) /

p0 3 → C factors via the reduction map C( Ép0 ) ( p0 ) modulo p0.

→ C Proof. Continuity means that g factors via the reduction C( Ép0 ) ( /p0 ) for

∈ n C a certain n Æ.The /p0 -valued points on reducing to the neutral element form a p0-group.

6.18. Proposition. Suppose that p0 =3 .Then,forα ∈ Br(X),thelocal evaluation map ev (α, ·) factors via

νp0 −→ red: X( Ép0 ) Ep( p0 )

(x0 : x1 : x2 : x3) → ((x0 mod p0):(x1 mod p0):(x2 mod p0)) . ∈ É

Proof. Let (x0 : x1 : x2 : x3) X( p0 ). We may assume without restriction that

∈ ∗ x0,x1,x2,x3 p0 are coprime. Then assumption ( ) implies that x0, x1,orx2 is a unit. Assume, again without restriction, that x2 is a unit. Then x0 and x1 cannot both be multiples of p0. Assume p0 x1.

Sec. 6] the “first case” of diagonal cubic surfaces 153

∈ ≡

Hensel’s lemma ensures that there exists a unique t p0 such that t x2 (mod p0) ∈ É and (x0 : x1 : t :0) X( p0 ). We claim that

ev (α, (x : x : x : x )) = ev (α, (x : x : t :0)). νp0 0 1 2 3 νp0 0 1

Proof of the claim: Both points are contained in the intersection of X with the hyperplane “x1x0 − x0x1 =0”. This is a genus one curve C.Itisgivenbythe equation 3 3 3 3 [a0(x0/x1) + a1]x1 + a2x2 + a3x3 =0.

The assumptions imply that the ﬁrst coeﬃcient is not divisible by p0.

As p0|a3,thecurveC has bad reduction at p0. The equation above deﬁnes a minimal

C É model of C over p. There are no p0 -valued points with singular reduction. C

p0 consists of three lines meeting in the singular point (0:0:0:1).Itmay happen that all three lines are deﬁned over p0 or that one line is deﬁned over p0

and the other two over p2 and are conjugate to each other. ∈ É We choose a basepoint x C( p0 ). Then, by Lemma 6.15,

· | É −→

evν (α, ) C( É ) : C( p ) / p0 p0 0 3 diﬀers from a continuous homomorphism of groups just by a constant sum- · |

mand. Lemma 6.17 shows that ev (α, ) É factors via the reduction map νp0 C( p0 )

reg → C C( Ép0 ) p0 ( p0 ).

reg

C C If only one line of p0 is deﬁned over p0 ,thenwehave# p0 ( p0 )=p0.Ev-

1 ery group homomorphism to 3 / is constant, which implies the claim in this case.

reg C Otherwise, we have # p0 ( p0 )=3p0, each line having exactly p0 smooth points. Let x be the third point of intersection of the line tangent to C in x with C. · Then the group structure on C( Ép0 ) has the property that P + Q + R =2 x + x if and only if P , Q,andR are collinear. In particular, P + Q is the third point on the line through R and the neutral element. C The reductions of three collinear points are either belonging to the same line of p0

reg C or to three diﬀerent lines. By consequence, the subgroup of p0 ( p0 ) of order p0 is provided by the line containing the neutral element. The three lines form its cosets.

As the reductions of (x0 : x1 : x2 : x3) and (x0 : x1 : t :0)are on the same line, this implies the claim. Now, we apply Theorem III.8.16 and Lemma 6.17 to the elliptic curve E given by x3 =0. This shows ev α, (x : x : t :0) = g (x mod p ):(x mod p ):(t mod p ) νp0 0 1 0 0 1 0 0

→ for a map g : Ep0 ( p0 ) 3 / .Since(t mod p0)=(x2 mod p0), the proof is complete.

6.19. Remark. In the case p0 =3 , the only assertion still to be proven is · |

that ev (α, ) É is non-constant. For that, according to Theorem III.8.16, it νp0 E( p0 )

154 an application: the brauer–manin obstruction [Chap. IV

∗ suﬃces to show that α|E ∈ Br(E)/π Br(Spec É) is non-zero. We will verify this in the next subsection in Proposition 6.24. The next lemma, although true in general, will be relevant for us only in the case p0 =3.

6.20. Lemma. Let p be an arbitrary prime number, and let α ∈ Br(X) be any Brauer class. Then there exists a positive integer n such that evνp (α, x)

(pn) n depends only on the reduction x ∈ X ( /p ).

Proof. This follows directly from the continuity of evνp with respect to the right argument. iii. Evaluating on the elliptic curve.

6.21. Notation. For the ﬁelds K and L as in Notation 6.11, we ﬁx a valuation ν of K lying above νp0 and an extension w of ν to L.

2 ∼

6.22. Lemma. Fix an isomorphism ι: H (σ, ) = /3 . Under the periodicity isomorphism

∼ H1(σ, Pic(X )) −→= Hˆ −1(σ, Pic(X )) Kν Kν ∩ 0 0 =(NDiv(XLw ) Div (XKν ))/N Div (XLw ) induced by ι, a representative of a generator is given by

1 1 f := (x0 +(a1/a0) 3 x1)/(x0 +(a2/a0) 3 x2) .

Proof. In fact,

div(f)=L2(0, 0) + L2(0, 1) + L2(0, 2) − L1(0, 0) − L1(0, 1) − L2(0, 2)

= N(L2(0, 0) − L1(0, 0)) .

Further, the norms of the principal divisors are generated by the pairwise diﬀerences of 1

NΔkn = Lk(0,i)+ Lk(1,i)+ Lk(2,i) ,

∈ ∈ ∈ i /3 i /3 i /3 2

NΔkm =3 Lk(m, i) ,

∈ i /3 3 −

NΔabc = L0(b a, i)+ L1(b, i)+ L2(a, i) .

∈ ∈ i∈/3 i /3 i /3

Ignoring the round brackets, these elements are 9Lk for k ∈{0, 1, 2} and 3L0 +3L1 +3L2. They do not generate 3L2 − 3L1.

6.23. Remark. The periodicity isomorphism is compatible with the action of the involution τ.Asf is τ-invariant, it represents a non-zero cohomology class in H1(σ, Pic(X ))τ . Kν

Sec. 6] the “first case” of diagonal cubic surfaces 155

6.24. Proposition. Assume p0 =3 . Then the pullback

∗ ∗

É −→ É

Br(X)/π Br(Spec ) Br(E É )/π Br(Spec ) p0 p0 is not the zero map. Proof. It suﬃces to show that the restriction

∗ −→ ∗ Br(X)/π Br(Spec É) Br(EKν )/π Br(Spec Kν ) is not the zero map. ∗ ∼ It factors via [Br(X )/π Br(Spec K )]τ = H1(σ, Pic(X ))τ ,forwhich,by Kν ν Kν Lemma 6.22, we have a generator f in explicit form.

It is, therefore, suﬃcient to show that there exists a point x ∈ E(Kν) such ∈ ∗ ∗ → ∗ that f(x) Kν is not in the image of the norm map N : Lw Kν . As Lw/Kν is totally ramiﬁed, this follows directly from the lemma below.

6.25. Lemma. Let p =3 be a prime number, let K be a ﬁnite ﬁeld exten-

sion of Ép,andletL/K be a totally ramiﬁed Galois extension of degree three. Further, let C be the genus one curve over K given by x3 + y3 + z3 =0. Then the range of the rational function f := (x + y)/(x + z) on C(K) contains an element that is not a norm under N : L∗ → K∗.

Proof. The residue field k := OK /mK is finite and #k ≡ 1(mod 3).Wehaveto show that the range of the rational function f =(x + y)/(x + z) on C(k) contains a non-cube in k∗. For this, we write P := ((−1) : 1 : 0) and Q := ((−1):0:1).Then div f =3(P ) − 3(Q). Choosing an arbitrary point O ∈ C(k) as the neutral element fixes a group law on C(k).Wehavethe3-division point P − Q on C. Consider the isogeny · i: C := C/P − Q −→3 C. √ i corresponds to the field extension k(C)=k(C) 3 f /k(C). We claim that i C(k) ∪{P, Q} = {x ∈ C(k) | f(x) ∈ (k∗)3}∪{P, Q} .

∈ Indeed, to give a point x C(k) is equivalent to giving a valuation νx : k(C) → ∼ such that, for the corresponding residue ﬁeld, one has Ox/mx = k. x belongs to the image of i(k) if and only if this valuation splits completely in k(C). Thus, our task is to show that i(k): C(k) → C(k) is not surjective. Note that C(k) − i contains at least the nine points given by (1 : ( ζ3):0)and permutations. The exact sequence of Gal(k/k)-modules

0 −→ μ3 −→ C (k) −→ C(k) −→ 0 induces a long exact sequence

C(k) −→ C(k) −→ k∗/(k∗)3 −→ 0 .

156 an application: the brauer–manin obstruction [Chap. IV

Observe, by Hasse’s bound, every genus one curve has a point overk.Thus,there are no non-trivial torsors over C and we have H1 Gal(k/k),C(k) =0. The assertion follows.

iv. Completing the proof for p0 =3. √

3 É 6.26. Observation. We have Kν = É3(ζ3) or Kν = 3(ζ3, 2). √ √

3 3 ∗ É Proof. By construction, Kν = É3(ζ3, a, b) for a, b units in 3. Modulo third ∗ powers, units√ in É3 fall into√ three classes, represented by 1, 2,and4, respectively.

3 3

É As É3(ζ3, 2) = 3(ζ3, 4), the assertion follows.

6.27. Notation. For F ∈ 3, we denote the genus one curve, given by the F 0 equation x3 = Fx0 on X,byE .Inparticular,E = E in our previous notation. F 3 3 3 The projection of E to the ﬁrst three coordinates is “Ax0 + a1x1 + a2x2 =0” 3 for A := a0 + F a3. The assumptions 3 a0 and 3|a3 make sure that A is a 3-adic unit. F F X We write E for the model of E over Spec 3 given by x3 = Fx0 on .

6.28. Proposition. For every F ∈ 3, the pullback

∗ F ∗ −→ É Br(X)/π Br(Spec É) Br(E )/π Br(Spec ) É3 3 is not the zero map. Proof. It suﬃces to show that the restriction

∗ −→ F ∗ Br(X)/π Br(Spec É) Br(EKν )/π Br(Spec Kν ) is not the zero map. ∗ ∼ It factors via [Br(X )/π Br(Spec K )]τ = H1(σ, Pic(X ))τ ,forwhich,by Kν ν Kν Lemma 6.22, we have a generator [f] in explicit form. F It is, therefore, suﬃcient to show that there exists a point x ∈ E (Kν ) such that ∈ ∗ ∗ → ∗ f(x) Kν is not in the image of the norm map N : Lw Kν .Sincea2/a1 is a ∗ cube in Kν , this follows directly from the lemma below.

∈ O 6.29. Lemma. Let A Kν be diﬀerent from zero, and let C be the 3 3 3 genus one curve over Kν given by Ax + y + z =0.

Then the range of the rational function f := (x + y)/(x + z) on C(Kν ) contains an ∗ → ∗ element that is not a norm under N : Lw Kν . Proof. First step. Description of the norms.

Let π be a uniformizing element of Kν . Then, according to [Se62, Chap. V, Propo- sition 5], there exists a positive integer t such that the homomorphism N :(1+mn )/(1 + mn+1) −→ 1+(π)n / 1+(π)n+1 n Lw Lw

Sec. 6] the “first case” of diagonal cubic surfaces 157 is a bijection for n

w ≡ 1+απt (mod πt+1) is not a norm. For t, there is the formula

νL(D ) t = Lw/Kν − 1, 2 where DLw/Kν denotes the diﬀerent of Lw/Kν . Second step. Calculation of the diﬀerent. √ 3

Case 1. K = É (ζ , 2). ν 3 √3 √

3 3 37 É Then Lw = 3(ζ3, 2, 3). For the discriminants, one calculates dLw/É3 =(3) 7 and dKν /É3 =(3) . This results in νL(DLw/Kν )=16and t =7.

For comparison, νK (3) = 6.

Case 2. K = É (ζ ). ν 3 √3 √ √

3 3 3

É É Then Lw = É3(ζ3, 3), 3(ζ3, 6),or 3(ζ3, 12). 11

For each possibility the discriminant is the same, dLw/É3 =(3) . Together with

dKν /É3 =(3), this shows νL(DLw/Kν )=8and t =3.

For comparison, νK (3) = 2.

Third step. Construction of the point. We claim that, on C,thereisaK-valued point (x : y : z) such that x = πt, y ≡−1/α (mod π),andz ≡ 1/α (mod π). Then the assertion follows, since x + y πt − 1/α απt − 1 ≡ ≡ ≡−(1 + απt)(mod πt+1) x + z πt +1/α απt +1 is not a norm. To show the existence of the point, we choose y ≡−1/α (mod π), arbitrarily. Then the equation g(Z):=Z3 +(y3 + Aπ3t)=0 has a solution Z ≡−y (mod π) by Hensel’s lemma since

3t νK (g(−y)) = νK (Aπ ) ≥ 3t, 2 νK (g (−y)) = νK (3y )=νK (3) , and 3t>2νK (3).

∗ ∈ É

6.30. Proposition. Assume that a2/a1 3 is a non-cube, and let ∈ Æ t =(t0 : t1 : t2 : t3) ∈ X ( 3) be a point with t0 =0. Choose n such

(pn) n

∈ X that evνp (α, x) depends only on the reduction x ( /p ).

158 an application: the brauer–manin obstruction [Chap. IV

Then, for any F ∈ 3 such that (F mod 3) = t3/t0, one has

{ ∈ E F n | } # y ( /3 ) y = t, evν3 (α, y)=0

{ ∈ E F n | } =# y ( /3 ) y = t, evν3 (α, y)=1/3

{ ∈ E F n | } =# y ( /3 ) y = t, evν3 (α, y)=2/3 .

F ∗

Proof. Proposition 6.28 shows that α| F ∈ Br(E )/π Br(Spec É ) is dif- E É 3

É 3 | 3 ferent from zero. Thus, α EF yields a surjective homomorphism of groups

F 1 É3

→ E ( É3) 3 / . From this, we immediately see

{ ∈ E F n F n | } { ∈ E | } # y ( /3 ) evν3 (α, y)=0 =#y ( /3 ) evν3 (α, y)=1/3

{ ∈ E F n | } =#y ( /3 ) evν3 (α, y)=2/3 .

We have to show the same for points reducing to t. ∈ ≡± A unit u 3 is a cube if and only if u 1(mod 9). Permuting coordinates and changing a sign, if necessary, we may therefore assume that a1 ≡ 1(mod 9) and a2 ≡ 7(mod 9). F 3 3 3 3 E is given by Ax0 + a1x1 + a2x2 =0for A := a0 + F a3. In principle, there are three cases. First case. A ≡±7(mod 9). ≡ F We may assume A 7(mod 9). Then, modulo 3,allÉ3-valued points on E reduce to (1 : 0 : (−1)). Hence, the assertion is true in this case. Second case. A ≡±4(mod 9). 3 3 3 3 This is impossible as 4x0+1x1+7x2 =0allows no solutions in É3 except for (0, 0, 0). Third case. A ≡±1(mod 9). ≡ F Assume without restriction that A 1(mod 9).AÉ3-valued point on E may reduce either to (1 : (−1) : 0) or to (1 : 1 : 1). F Take (1 : (−a):0)∈ E ( É3),fora the third root of A/a1, as the neutral element.

F F → E Then Lemma 6.31 shows that the reduction map E ( É3) ( 3) gives rise to

→ a surjective group homomorphism E ( É3) /2 . · As 2 is prime relative to 3, the asserted equidistribution of evν3 (α, ) holds in each class, separately.

6.31. Lemma. Consider the elliptic curve C over É3, given by

x3 + y3 +7z3 =0.

Take (1 : (−1) : 0) as its basepoint.

C has a minimal Weierstraß model such that C0( É3), the group of points with non-singular reduction, coincides with the set of the points reducing (naively)

− ∼ É to (1 : ( 1) : 0).Further,C( É3)/C0( 3) = /2 . Proof. The substitutions 63 x := −21z, y := (x − y) ,z := x + y 2

Sec. 6] the “first case” of diagonal cubic surfaces 159 lead to the Weierstraß equation 7233 zy2 = x3 − z3. 4

If the reduction of (x : y : z) is (1:1:1),then(x : y : z) reduces to (0:0:1). This is the cusp. On the other hand, consider the case that x =1, z =3k, and, therefore, y ≡−1 − 7 · 32k3 (mod 27). Dividing the substitution formulas by 63, we obtain − ≡ 7 2 3 ≡ − 3 x = k, y 1+ 2 3 k (mod 27), i.e., y 1(mod 9),andz = k (mod 3). In particular, y is always a unit. The reduction of (x : y : z) is never equal to the cusp.

∼

É It remains to show that C( É3)/C0( 3) = /2 and that the Weierstraß model found is minimal. For this, we follow Tate’s algorithm [Ta75]. We have the aﬃne equation 7233 Y 2 = X3 − . 4 According to [Ta75, Summary], we are in case 6). The polynomial P is given 3 − 72 by P (T )=T 4 . It has a triple zero modulo 3. 72 ≡ 3 72 We observe that 4 1(mod 9) and write u for the 3-adic unit such that u = 4 . The substitution X := X − 3u leads to Y 2 = X3 +9uX2 +27u2X. Tate’s equation (8.1) becomes 2 3 2 2 Y2 =9X2 +9uX2 +3u X2 .

Here, Y =: 9Y2 and X =: 9X2. Equation (9.1) is

2 3 2 2 3Y3 =3X2 +3uX2 + u X2

2 for Y2 =: 3Y3.Asu is a unit, we see that the Weierstraß model is minimal and of

∗ ∼

É Kodaira type III .[Ta75, table on p. 46] indicates C( É3)/C0( 3) = /2 .

6.32. Proposition. Let X be any diagonal cubic surface fulﬁlling (∗). ∈ Choose n Æ such that the evaluation evνp (α, x) depends only on the reduction

(pn) n x ∈ X ( /p ).

Then, for every t =(t0 : t1 : t2 : t3) ∈ X ( 3), we have

| } { ∈ X # y ( /3 ) y = t, evν3 (α, y)=0

{ ∈ X n | } =# y ( /3 ) y = t, evν3 (α, y)=1/3

{ ∈ X n | } =# y ( /3 ) y = t, evν3 (α, y)=2/3 .

Proof. The case a0 ≡ a1 ≡ a2 (mod 9) is treated in Example 6.33.

We may therefore assume that two of the quotients a0/a1, a1/a2,anda2/a0, ∗ say a1/a2 and a2/a0, are non-cubes in É3. It is impossible that t0 = t1 =0. Again without restriction, assume t0 =0 .

160 an application: the brauer–manin obstruction [Chap. IV

We consider the projection

π : X −→ P1 ,

(x0 : x1 : x2 : x3) → (x3 : x0) .

On a É3-valued point x reducing to t, the birational map π is deﬁned. ∈ We have π(x)=(F :1)for some F 3 such that (F mod 3) = t3/t0. By Proposition 6.30, the asserted equality is true in every ﬁber of π. It is therefore

true in general. É 6.33. Example. Let d ∈ , and consider the cubic surface over given by

3 3 3 3 x0 + x1 + x2 +3dx3 =0.

By the results shown, we have

∗ 1 ∼

Br(X)/π Br(Spec É)=H Gal(L/K), Pic(XL) = /3 √ 3 for L = É(ζ3, 3d), which is equal to the ﬁeld of deﬁnition of the 27 lines on X.

As K = É(ζ3) does not contain any cubic extensions, the Brauer–Manin obstruction may be described completely explicitly, without relying on Lichtenbaum’s duality. In fact, 1 ∼ 1 τ H Gal(L/É), Pic(XL) = H σ , Pic(XL)

∗ τ É =[Br(X É(ζ3))/π Br(Spec (ζ3))] .

For the latter, we have the explicit generator [f] for f := (x0 + x1)/(x0 + x2). ∈ É This means, for x =(x0 : x1 : x2 : x3) ∈ X( É3(ζ3)) and ν Val( (ζ3)) the extension of ν3,wehave evν ([f],x)=iν (x0 + x1)/(x0 + x2) .

Here, iν is the homomorphism ∼ ∼

∗ ∗ ∗ = ˆ 0 ∗ = 2 ∗ invν

−→ É −→ −→ −→ É É3(ζ3) 3(ζ3) /N Lw H ( σ ,Lw) H ( σ ,Lw) /

for w the extension of ν to L. É However, we want to consider É3-valued points, not 3(ζ3)-valued ones. Thus, let ∈ ∈ α Br(X) be a Brauer class mapping to [f] under restriction. Then, for x X( É3),

· 2 evν3 (α, x)=evν ([f],x) .

1 Fortunately, multiplication by 2 is an automorphism of 3 / .

On X, two kinds of É3-valued points may be distinguished.

Sec. 7] concluding remark 161

First kind. x0, x1,andx2 are units.

Then x0 ≡ x1 ≡ x2 (mod 3) and d 3.Wehave ⇐⇒ ≡ evν3 (α, x)=0 x1 x2 (mod 9) .

Second kind. Among x0, x1,andx2, there is an element that is a multiple of 3.

Suppose 3|x0.Thenx1 ≡−x2 (mod 3).Wehave ⇐⇒ evν3 (α, x)=0 2x0 + x1 + x2 (mod 9) .

3 3 3 3 ∈ 6.34. Corollary. If x0 + x1 + x2 +3x3 =0,forx0, ... ,x3 ,thenwe have the following non-trivial congruences. i) If x0, x1,andx2 are units, then

x0 ≡ x1 ≡ x2 (mod 9) . ii) If 3|x0,then x0 ≡ 0(mod 9) . √ 3 Proof. This is the particular case d =1.ThenL = É(ζ3, 3) is unramiﬁed at every prime diﬀerent from 3.Asνp (x0 + x1)/(x0 + x2) is always divisible by 3, we have evνp (α, x)=0for every p =3. The congruences obtained from evν3 (α, x)=0imply the congruences asserted.

6.35. Remark. Thus, we have shown that, for the equation

3 3 3 3 x0 + x1 + x2 +3x3 =0

over É, the restrictions coming from the Brauer–Manin obstruction are exactly the congruences obtained by D. R. Heath-Brown in [H-B92a]. Note that Heath-Brown’s approach is more elementary. His main tool is the law of cubic reciprocity.

7. Concluding remark

7.1. Remark (Other examples). The literature on the algebraic Brauer– Manin obstruction is growing rapidly. Many more classes of examples have been studied. It is impossible to mention all of them. There is, however, one class that should be mentioned, namely Del Pezzo surfaces of degree 4. Here, M. Bright and his coworkers [B/B/F/L] understand the Brauer–Manin obstruction in such an explicit manner that it could be realized as an algorithm in magma.

Part C

Numerical experiments

CHAPTER V

The Diophantine equation x4 +2y4 = z4 +4w4∗

Hash, x. There is no deﬁnition for this word—nobody knows what hash is. Ambrose Bierce: The Devil’s Dictionary (1906)

Numerical experiments and the Manin conjecture

Part C of this book is devoted to experiments related to the conjecture of Manin, and in the reﬁned form due to E. Peyre. It mainly consists of two chapters, each of which presents the investigations on one or two particular samples of varieties. In addition, there is the present chapter, which is introductory. The experiments we are going to report about were carried out by Andreas-Stephan Elsenhans together with the author. Our selection of subjects was, of course, arbitrary to a certain extent. We do not claim it was mandatory in any sense to consider exactly the samples we considered. Nor do we want to give the reader the impression that nobody else ever made experiments related to the Manin conjecture. To the contrary, important experiments have been accomplished, for instance, by D. R. Heath-Brown [H-B92a]. He studied cubic surfaces where weak approximation fails. Heath-Brown’s investigations brought to light the fact that the failure of weak approximation is no reason to expect a slower growth of the number of

É-rational points. The main term from the circle method, carried over in the most naive manner that one can think of, seemed to ﬁt perfectly well in the examples, presented in [H-B92a]. Cf. Remark II.7.5.iv) in the ﬁrst part of the book. Later on, E. Peyre and Y. Tschinkel [Pe/T] dealt with the situation in which there is a non-trivial Brauer group, but the Brauer–Manin obstruction does not exclude any adelic point. Their experiments demonstrated that on such Fano varieties

(cubic surfaces, actually, in their experiments), there are more É-rational points than naively expected, in their case well by a factor of three. This showed that any serious deﬁnition of what today is called Peyre’s constant must take the factor β into account.

In general, experimenting on Manin’s conjecture includes calculating Peyre’s con-

stant and searching for É-rational points. To calculate Peyre’s constant, one may

∗This chapter collects material from the articles: The Diophantine equation x4 +2y4 = z4 +4w4, Math. Comp. 75 (2006), 935–940; and The Diophantine equation x4 +2y4 = z4 +4w4—a number of improvements, Preprint, both with A.-S. Elsenhans and the author.

165

166 the diophantine equation x4 +2y4 = z4 +4w4 [Chap. V essentially proceed according to the definition. The most interesting point is certainly the computation of the special value of the L-function at 1. But even for this, the main ideas go back at least to the 1960s, cf. Remark VII.6.6 below. On the other hand, algorithmic resolution of Diophantine equations is a very interesting and active field. To get an overview of the state of the art in the year 1998, we refer to the book of N. P. Smart [Sm]. Further, M. J. Bright’s experiments with diagonal quartic surfaces [Bri02] are of significant interest, albeit in this case the relevant arithmetic conjectures are wide open.

The present chapter is devoted to the Diophantine equation x4 +2y4 = z4 +4w4. As this defines a K3 surface, the conjecture of Batyrev-Manin would apply and, in fact, all the major questions concerning the arithmetic of this surface are open. However, the extreme sparsity of the solutions prevents us even from making an educated guess concerning their asymptotics. In fact, the goal of Chapter V is entirely different. The truth is, we use that particular equation mainly in order to present in detail our algorithm for the resolution of Diophantine equations. This algorithm works in general for decoupled equations. In this situation, it runs significantly faster than the methods applicable in general. It is related to the algorithm of Daniel Bernstein [Be].

The two main chapters of this part describe our experiments related to the Manin conjecture for some particular samples of varieties, diagonal cubic and quartic threefolds, and diagonal cubic surfaces. As all these are given by decoupled equations,

we may apply our algorithm to eﬃciently search for É-rational points. Generally speaking, Part C is most likely easier to follow than the others, in particular for all those readers who are well acquainted with computers and the concept of an algorithm. Let us emphasize, for example, that Chapter V does not require any of the advanced prerequisites listed on the very ﬁrst pages of this book. Fur- ther, each of the chapters starts with an introduction that tries to make it accessible even for a reader who did not study Parts A and B very well.

1. Introduction

1.1. Chapter II was devoted to very general conjectures on rational points on algebraic varieties. Let us brieﬂy recall a few facts.

i) An algebraic curve C of genus g>1 over É admits at most a ﬁnite number É of É-rational points. On the other hand, for genus one curves, #C( ) may be

zero, ﬁnite non-zero, or inﬁnite. For genus zero curves, one automatically has

∞ É ∅ #C( É)= as soon as C( ) = . ii) In higher dimensions, there is a conjecture, due to S. Lang, stating that if X is a variety of general type over a number ﬁeld, then all but ﬁnitely many of its rational points are contained in the union of closed subvarieties that are not of general type (cf. Conjecture II.2.2). On the other hand, abelian varieties (as well

as, e.g., elliptic and bielliptic surfaces) behave like genus one curves. I.e., #X( É) may be zero, ﬁnite non-zero, or inﬁnite. Finally, rational and ruled varieties comport in the same way as curves of genus zero in this respect.

Sec. 2] congruences 167

This list does not yet exhaust the classiﬁcation of algebraic surfaces, to say nothing of dimension three or higher. In particular, the following problem is still open.

1.2. Problem. Does there exist a K3 surface X over É that has a ﬁnite

É ∞ non-zero number of É-rational points? I.e., such that 0 < #X( ) < ?

1.3. Remark. This question was posed by Sir Peter Swinnerton-Dyer as Problem/Question 6.a) in the problem session to the workshop [Poo/T]. We are not able to give an answer to it.

1.4. One possible candidate for a K3 surface with the property ∞ 0 < #X( É) < is given by the following. Problem. Find a third point on the projective surface X ⊂ P3 deﬁned by

x4 +2y4 = z4 +4w4.

1.5. Remarks. i) The Problem in Subsection 1.4 is also due to Sir Pe- ter Swinnerton-Dyer [Poo/T, Problem/Question 6.c)]. It was raised, in particular, during his talk [SD04, very end of the article] at the Göttingen Mathematisches Institut on June 2, 2004. ii) x4 +2y4 = z4 +4w4 is a homogeneous quartic equation. It, therefore, deﬁnes a K3 surface X in P3. As trivial solutions of the equation, we consider those − corresponding to the É-rational points (1:0:1:0) and (1:0:( 1):0). iii) Our main result is the following theorem, which contains an answer to Prob- lem 1.4.

1.6. Theorem. The diagonal quartic surface X in P3 given by

x4 +2y4 = z4 +4w4 (∗)

admits precisely ten É-rational points having integral coordinates within the hypercube |x|, |y|, |z|, |w| < 108. These are (±1:0:±1:0) and (±1 484 801:±1 203 120:±1 169 407:±1 157 520).

1.7. Remark. This result clearly does not exclude the possibility that

#X( É) is actually ﬁnite. It might indicate, however, that a proof for this property is deeper than one originally hoped for.

2. Congruences

2.1. It seems natural to ﬁrst try to understand the congruences

x4 +2y4 ≡ z4 +4w4 (mod p) (†) modulo some prime number p.Forp =2and 5, one ﬁnds that all primitive solutions

168 the diophantine equation x4 +2y4 = z4 +4w4 [Chap. V

in satisfy a) x and z are odd, b) y and w are even, c) y is divisible by 5.

For other primes, it follows from the Weil conjectures, proven by P. Deligne [Del], that the number of solutions of the congruence (†)is

− 2 3 − #CX( p)=1+(p 1)(p + p +1+E)=p + E(p 1) .

Here, E is an error-term, which may be estimated by |E|≤21p. ∗ Indeed, consider the projective variety X over É deﬁned by ( ). It has good reduction at every prime p =2 . Therefore, [Del, Théorème (8.1)] may be applied 2 | |≤ to the reduction Xp. This yields #Xp( p)=p +p+1+E and E 21p. We note 2 X that dim H (X , Ê)=22for every complex surface of type K3 [Bv, p. 98].

2.2. Another question of interest is to count the numbers of solutions to the 4 4 4 4 congruences x +2y ≡ c (mod p) and z +4w ≡ c (mod p) for a certain c ∈ .

l This means that we count the p-rational points on the aﬃne plane curves Cc r 4 4 4 4 and Cc deﬁned over p by x +2y = c and z +4w = c, respectively. If p c and p =2 , then these are smooth curves of genus three.

By the work of André Weil [We48, Corollaire 3 du Théorème 13], the numbers of

p-rational points on their projectivizations are given by

l r #Cc( p)=p +1+El and #Cc( p)=p +1+Er, where the error-terms can be bounded by |El|, |Er|≤6 p.Theremaybeupto

four p-rational points on the inﬁnite line. For our purposes, it suﬃces to notice that both congruences admit a number of solutions that is close to p.

The case p|c, p =2 , is slightly diﬀerent since it corresponds to the case of a reducible curve. The congruence x4 + ky4 ≡ 0(mod p) admits only the trivial solution if (−k) is not a biquadratic residue modulo p. Otherwise, it has exactly 1+(p − 1) gcd(p − 1, 4) solutions.

l l r r

 Finally, if p =2,then#C0( 2)=#C1( 2)=#C0 ( 2)=#C1 ( 2)=2.

2.3. Remark. The number of solutions of the congruence (†)is

l r

 · #CX( p)= #Cc( p) #Cc ( p).

c∈p Hence, the formulas just mentioned yield an elementary estimate for that count. They show once more that the dominating term is p3. The estimate for the error is, however, less sharp than the one obtained via the more sophisticated methods in 2.1.

Sec. 4] an algorithm to efficiently search for solutions 169

3. Naive methods

3.1. The most naive method to search for solutions of (∗) is probably the following. Start with the set

| ≤ ≤ } {(x, y, z, w) ∈ 0 x, y, z, w N and test the equation for every quadruple. Obviously this method requires about N 4 steps. It can be accelerated using the congruence conditions for primitive solutions noticed above.

3.2. A somewhat better method is to start with the set

4 4 4 ≤ ≤ } {x +2y − 4w | x, y, w ∈ , 0 x, y, w N and to search for fourth powers. This set has about N 3 elements, and the algorithm takes about N 3 steps. Again, it can be sped up by the above congruence conditions for primitive solutions. We used this approach for a trial run with N =104. An interesting aspect of this algorithm is the optimization by further congruences. Suppose x and y are ﬁxed. Then about one half or three-quarter of the values for w are no solutions to the congruence modulo a new prime. Following this way, one can ﬁnd more congruences for w and the size of the set may be reduced by a constant factor.

4. An algorithm to eﬃciently search for solutions i. The basic idea.

4.1. We need to compute the intersection of two sets

4 4 4 4

≤ ≤ }∩{ | ∈ ≤ ≤ } {x +2y | x, y ∈ , 0 x, y N z +4w z,w , 0 z,w N .

Both have about N 2 elements. It is a standard problem in computer science to find the intersection of two sets that both fit into memory. Using the congruence conditions modulo 2 and 5,one can reduce the size of the first set by a factor of 20 and the size of the second set by a factor of 4. ii. Some details.

4.2. The two sets described above are too big, at least for our computers and interesting values of N. Therefore, we introduced a prime number pp,which we call the page prime. Deﬁne the sets

4 4 ≤ ≤ 4 4 ≡ } Lc := {x +2y | x, y ∈ , 0 x, y N,x +2y c (mod pp)

170 the diophantine equation x4 +2y4 = z4 +4w4 [Chap. V and

4 4 ≤ ≤ 4 4 ≡ } Rc := {z +4w | z,w ∈ , 0 z,w N,z +4w c (mod pp) .

This means the intersection problem is divided into pp pieces and the sets Lc and Rc 6 ﬁt into the computer’s memory if pp is big enough. We worked with N =2.5 · 10 and chose pp = 30 011.

For every value of c, our program computes Lc and stores this set in a hash table. Then it determines the elements of Rc and looks them up in the table. Assum- ing uniform hashing, the expected running-time of this algorithm is O(N 2).

4.3. Remark An important further aspect of this approach is that the problem may be attacked in parallel on several machines. The calculations for one particular value of c are independent of the analogous calculations for another one. Thus, it is possible, say, to let c run from 0 to (pp − 1)/2 on one machine and, at the same time, from (pp +1)/2 to (pp − 1) on another. iii. Some more details.

4.4. (The page prime). For each value of c, it is necessary to ﬁnd the solu- 4 4 4 4 tions of the congruences x +2y ≡ c (mod pp) and z +4w ≡ c (mod pp) in an eﬃcient manner. We do this in a rather naive way by letting y (w) run from 0 4 4 to pp − 1. For each value of y (w), we compute x (z ). Then we extract the fourth root modulo pp.

Note that the page prime fulﬁlls pp ≡ 3(mod 4). Hence, the fourth roots of unity modulo p are just ±1 and, therefore, a fourth root modulo pp, if it exists, is unique up to sign. This makes the algorithm easier to implement.

4.5. Actually, we do not execute any modular powering operation or even computation of fourth roots in the lion’s share of the running-time. For more eﬃciency, all fourth powers and all fourth roots modulo pp are computed and stored in an array during an initialization step. Thus, the main speed limitation to ﬁnd all solutions to a congruence modulo pp is, in fact, the time it takes to look up values stored in the machine’s main memory.

4.6. (Hashing). We do not compute Lc and Rc directly, because this would require the use of multiprecision integers within the inner loop. Instead, we choose two other primes, the hash prime ph and the control prime pc, which ﬁt into the 32-bit registers of our computers. All computations are done modulo ph and pc. More precisely, for each pair (x, y) considered, the expression 4 4 (x +2y ) mod ph deﬁnes its position in the hash table. In other words, we hash pairs (x, y) whereas → 4 4 (x, y) (x +2y ) mod ph plays the role of the hash function. For each pair (x, y), 4 4 we write two entries into the hash table, namely the value of (x +2y ) mod pc and the value of y.

Sec. 5] general formulation of the method 171

In the main computation, we worked with the numbers ph = 25 000 009 for the hash prime and pc = 400 000 009 for the control prime.

4.7. Note that, when working with a particular value of c, there are around pp pairs ((x mod pp), (y mod pp)) that fulﬁll the required congruence

4 4 x +2y ≡ c (mod pp) .

Therefore, approximately N/2 N/10 N 2 pp · · = pp pp 20pp

2 values will be written into the table. For our choices, N ≈ 10 412 849,which 20pp means that the hash table will get approximately 41.7% ﬁlled.

As for many other rules, there is an exception to this one. If c =0, then approximately 1+(pp − 1) gcd(pp − 1, 4) pairs ((x mod pp), (y mod pp)) may satisfy the congruence 4 4 x +2y ≡ 0(mod pp) .

As pp ≡ 3(mod 4) this is not more than 2pp − 1, and the hash table will be ﬁlled not more than about 83.3%.

4.8. To resolve collisions within the hash table, we use an open addressing method. We are not particularly afraid of clustering and choose linear probing. We feel free to use open addressing as, thanks to the Weil conjectures, we have a priori estimates available for the load factor.

4.9. The program makes frequent use of fourth powers modulo ph and pc. Again, we compute these data in the initialization part of our program and store them in arrays, once and for all.

5. General formulation of the method

5.1. The method described in the previous section is actually a systematic method to search for solutions of a Diophantine equation. It works eﬃciently when the equation is of the form

f(x1,...,xn)=g(y1,...,ym) .

We ﬁnd all solutions that are contained within the (n + m)-dimensional cube

{ n+m || | | |≤ } (x1,...,xn,y1,...,ym) ∈ xi , yi B .

The expected running-time of the algorithm is O(Bmax{n,m}).

172 the diophantine equation x4 +2y4 = z4 +4w4 [Chap. V

5.2. The basic idea may be formulated as follows.

Algorithm H. i) Evaluate f on all points of the n-dimensional cube

{ n || |≤ } (x1,...,xn) ∈ xi B .

Store the values within a set L. ii) Evaluate g on all points of the cube

{ m || |≤ } (y1,...,ym) ∈ yi B of dimension m. For each value start a search in order to ﬁnd out whether it occurs in L. When a coincidence is detected, reconstruct the corresponding values of x1,...,xn and output the solution.

5.3. Remarks. a) In fact, we are interested in the very particular Diophan- tine equation x4 +2y4 = z4 +4w4 , which was suggested by Sir Peter Swinnerton-Dyer. b) i) In the form stated above, the main disadvantage of Algorithm H is that it requires an enormous amount of memory. Actually, the set L is too big to be stored in the main memory even of our biggest computers, already when the value of B is only moderately large.

For that reason, we introduced the idea of paging.Wechooseapage prime pp and work with the sets Lr := {s ∈ L | s ≡ r (mod pp)} for r =0,...,pp −1, separately. At the cost of some more time spent on initializations, this yields a reduction of the memory space required by a factor of 1 . pp ii) The sets Lr were implemented in the form of a hash table with open addressing. iii) It is possible to achieve a further reduction of the running-time and the memory required by making use of some obvious congruence conditions modulo 2 and 5.

5.4. The goal of the remainder of this chapter is to describe an improved implementation of Algorithm H, which we used in order to ﬁnd all solutions of x4 +2y4 = z4 +4w4, contained within the hypercube { ∈ 4 || | | | | | | |≤ 8} (x, y, z, w) x , y , z , w 10 .

6. Improvements I—more congruences

6.1. The most obvious way to further reduce the size of the sets Lr and to increase the speed of Algorithm H is to ﬁnd further congruence conditions for solutions and evaluate f and g only on points satisfying these conditions. As the equation we are interested in is homogeneous, it is suﬃcient to restrict consideration to primitive solutions.

Sec. 6] improvements i—more congruences 173

6.2. It should be noticed, however, that this idea is subject to strict limitations. If we were using the most naive O(Bn+m)-algorithm, then, for more or ∈ ≡ less every l Æ, the congruence f(x1, ... ,xn) g(y1, ... ,ym)(mod l) caused a reduction of the number of (n+m)-tuples to be checked. For Algorithm H, however, the situation is by far less fortunate. One may gain something only if there are residue classes (r mod l) that are represented by f, but not by g, or vice versa. Values, the residue class of which is not represented by g, do not need to be stored into Lr. Values, the residue class of which is not represented by f, do not need to be searched for. Unfortunately, if l is prime and not very small, then the Weil conjectures ensure that all residue classes modulo l are represented by both f and g.Inthiscase, the idea fails completely. The same is, however, not true for prime powers l = pk. ∂f Hensel’s lemma does not work when all partial derivatives ∂x (x1,...,xn), respec- ∂g i tively (y1,...,ym), are divisible by p. This makes it possible that certain residue ∂yi classes (r mod pk) are not representable although (r mod p) is. i. The prime 5. Congruences modulo 625.

6.3. In the algorithm described in the previous chapter, we made use of the fact that y is always divisible by 5. However, at this point, one can do a lot better. 4

When one takes into consideration that a ≡ 1(mod 5) for every a ∈ not divisible by 5, a systematic inspection shows that there are actually two cases. Either, 5|w.Then5x and 5z. Or, otherwise, 5|x.Then5z and 5w.Notethat, in the latter case, one indeed has z4 +4w4 ≡ 1+4≡ 0(mod 5).

6.4. The case 5|w. We call this case “N” and use the letter N in a prominent position in the naming of the relevant ﬁles of source code. N stands for “normal”. Considering this case as the ordinary one is justiﬁed by the fact that all primitive solutions known actually belong to it. Note, however, that we have no theoretical reason to believe that this case should in whatever sense be better than the other one.

In case N, we rearrange the equation to fN (x, z)=gN (y, w),where

4 4 4 4 fN (x, z):=x − z and gN (y, w):=4w − 2y .

As y and w are both divisible by 5,weget

4 4 gN (y, w)=4w − 2y ≡ 0(mod 625).

Consequently, fN (x, z) ≡ 0(mod 625).

This yields an enormous reduction of the set Lr. To see this, recall 5x and 5z. That means for x there are precisely ϕ(625) possibilities in /625 .Further,for each such value, the congruence z4 ≡ x4 (mod 625) may not have more than four so-

· ∈ 2 lutions. All in all, there are 4 ϕ(625) = 2 000 possible pairs (x, z) ( /625 ) . Further, these pairs are very easy to ﬁnd, computationally. The fourth roots of unity

± ± ∈ ∗ modulo 625 are 1 and 182.Foreachx /625 , put z := (±x mod 625) and z := (±182x mod 625).

174 the diophantine equation x4 +2y4 = z4 +4w4 [Chap. V

2 We store the values of fN into the set Lr.Only2 000 out of 625 values (0.512%) need to be computed and stored. Then each value of gN is looked up in Lr. Here, as y and w are both divisible by 5, only one value out of 25 (4%) needs to be computed and searched for.

6.5. The case 5|x. We call this case “S” and use the letter S in a prominent position in the naming of the relevant ﬁles of source code. S stands for “Sonderfall”, which means “exceptional case”. It is not known whether there exists a solution belonging to case S. Here, we simply interchange both sides of the equation. Deﬁne

4 4 4 4 fS(z,w):=z +4w and gS(x, y):=x +2y .

As x and y are divisible by 5,wegetx4 +2y4 ≡ 0(mod 625) and, therefore, z4 +4w4 ≡ 0(mod 625). Again, this congruence allows only 4 · ϕ(625) = 2 000 solutions

∈ 2 (z,w) ( /625 ) ,

− and these pairs are easily computable, too. The fourth roots of ( 4) in /625 are

± ± ∈ ∗ 181 and 183.Foreachx /625 , one has to consider z := (±181x mod 625) and z := (±183x mod 625).

We store the values of fS into the set Lr. Then we search through Lr for the values 2 of gS .Asabove,only2 000 out of 625 values need to be computed and stored, and one value out of 25 needs to be computed and searched for. ii. The prime 2.

6.6. Any primitive solution is of the form that x and z are odd while y and w are even.

6.7. In case S, there is no way to do better than that as both fS and gS represent (r mod 2k) for k ≥ 4 if and only if r ≡ 1(mod 16). 4 4 In case N, the situation is somewhat better. gN (y, w)=4w −2y is always divisible 4 4 by 32 while fN (x, z)=x − z ≡ 0(mod 32), as may be seen by inspecting the fourth roots of unity modulo 32, implies the condition x ≡±z (mod 8).Thismay be used to halve the size of Lr. iii. The prime 3.

6.8. Looking for further congruence conditions, a primitive solution must necessarily satisfy, we did not ﬁnd any reason to distinguish more cases. But there are a few more congruences, which we used in order to reduce the size of the sets Lr. To explain them, let us ﬁrst note two theorems on binary quadratic forms. They may both be easily deduced from [H/W, Theorems 246 and 247].

Sec. 6] improvements i—more congruences 175

2 2 2 2 6.9. Theorem. The quadratic forms q1(a, b):=a + b , q2(a, b):=a − 2b , 2 2 and q3(a, b):=a +2b admit the property below.

Suppose n0 := qi(a0,b0) is divisible by a prime p that is not represented by qi. Then p|a0 and p|b0.

6.10. Theorem. A prime number p is represented by q1, q2,orq3, respectively, if and only if (0 mod p) is represented in a non-trivial way. In particular, ≡ i) p is represented by q1 if and only if p =2or p 1( mod 4). 2 ii) p is represented by q2 if and only if p =2or p =1. The latter means ≡ p 1, 7(mod 8). −2 iii) p is represented by q3 if and only if p =2or p =1. The latter is equivalent to p ≡ 1, 3(mod 8).

6.11. Remark There is the obvious asymptotic estimate

{ n ∈ È ≤ }∼ q (a, b) | a, b ∈ ,q (a, b) ,q (a, b) n . i i i 2logn Further,

{ | |≤ }∼ n qi(a, b) | a, b ∈ , qi(a, b) n Ci log n where C1, C2,andC3 are constants, which can be expressed explicitly by Eu- ler products. (For q1, this is worked out in [Brü, Satz (1.8.2)]. For the other forms, J. Brüdern’s argument works in the same way without essential changes.)

6.12. Congruences modulo 81. In case N,

2 2 2 2 2 2 gN (y, w)=(2w ) − 2(y ) = q2(2w ,y ) ,

If 3|fN (x, z) but 81fN (x, z),thenfN (x, z) does not need to be stored into Lr. Further, if 3|x and 3|z,thenfN (x, z) does not need to be stored, either, as it cannot lead to a primitive solution. This reduces the size of the set Lr by a factor 1 · 1 1 − 1 131 ≈ of 9 +4 3 ( 3 81 )= 243 53.9%. In case S, the situation is the other way around.

2 2 2 2 2 2 fS (z,w)=(z ) +(2w ) = q1(z , 2w )

2 2 and q1 does not represent the prime 3. Therefore, if 3|fS(z,w),then3|z and 3|2w , which implies that z and w are both divisible by 3 and 81|fS (z,w).

We use this in order to reduce the time spent on reading. If 3|gS(x, y) but 81gS(x, y) or if 3|x and 3|y,thengS(x, y) does not need to be searched for. Although modular operations are not at all fast, the reduction of the number of attempts to read by 53.9% is highly noticeable.

176 the diophantine equation x4 +2y4 = z4 +4w4 [Chap. V iv. Some more hypothetical improvements.

6.13. i) In the argument for case N given above, p =3might be replaced by any other prime p ≡ 3, 5(mod 8). In case S, the same argument as above works for every prime p ≡ 3(mod 8). For primes p ≡ 5(mod 8), the strategy could be reversed. q3 is a binary quadratic form that represents (0 mod p) only in the trivial manner. Therefore, if p|gS(x, y), then p|x and p|y. It is unnecessary to store fS (z,w) if p|z and p|w or if p|fS(z,w) 4 but p fS (z,w). i) Each argument mentioned may be extended to some primes p ≡ 1(mod 8). For example, in case N, what is actually needed is that 2 is not a fourth power modulo p.Thisistrue,e.g.,forp =17, 41,and97, but not for p =73and 89. ii) fN and fS do not represent the residue classes of 6, 7, 10,and11 modulo 17. gN and (−gS) do not represent 1, 3,and9 modulo 13. This could be used to reduce the load for writing as well as reading.

6.14. Remarks. a) We did not implement these improvements as it seems the gains would be marginal or the cost of additional computations would even dominate the eﬀect. It is, however, foreseeable that these congruences will eventu- ally become valuable when the speed of the CPUs available will continue to grow faster than the speed of memory. Observe that alone the congruences noticed in a) could reduce the amount of data to be stored into L to a size asymptotically less than εB2 for any ε>0. b) For every prime p diﬀerent from 2, 5, 13,and17, the quartic forms fN , gN , fS,andgS represent all residue classes modulo p. This means that ii) may not be carried over to any further primes.

This can be seen as follows. Let b be equal to fN , fS, gN ,orgS. (0 mod p) is represented by b, trivially. Otherwise, b(x, y)=r defines an affine curve Cr of genus three with at most four points on the infinite line. The Weil conjectures [We48, Corollaire 3 du Théorème 13] imply that [(p+1−6 p)−4] is a lower bound for the ≥ number of p-rational points on Cr. This is a positive number as soon as p 43. In this case, every residue class (r mod p) is represented, at least, once. For the remaining primes up to p =41, an experiment shows that all residue classes modulo p are represented by fN , fS , gN ,aswellasgS.

7. Improvements II—adaptation to our hardware i. A 64-bit based implementation of the algorithm.

7.1. We migrated the implementation of Algorithm H from a 32-bit processor to a 64-bit processor. This means the new hardware supports addition and multiplication of 64-bit integers. Even more, every operation on (unsigned) integers is automatically modulo 264. From this, various optimizations of the implementation described in Section 4 are almost compelling. The basic idea is that 64 bits should be enough to deﬁne hash value and control value by selection of bits instead of using (notoriously slow)

Sec. 7] improvements ii—adaptation to our hardware 177 modular operations. Hash value and control value are two integers signiﬁcantly less than 232, which should be independent on each other. Note, however, that the congruence conditions modulo 2 imposed imply that x4 ≡ z4 ≡ 1(mod 16) and 2y4 ≡ 4w4 ≡ 0(mod 16). This means, the four least signiﬁcant bits of f and g may not be used as they are always the same.

7.2. The description of the algorithm below is based on case S, case N being completely analogous. Algorithm H64. I. Initialization. Fix B := 108. Initialize a hash table of 227 = 134 217 728 integers, each being 32 bits long. Fix the page prime pp := 200 003. Further, deﬁne two functions, the hash function h and the control function c,which map 64-bit integers to 27-bit integers and 31-bit integers, respectively, by selecting certain bits. Do not use any of the bits twice to ensure h and c are independent on each other and do not use the four least signiﬁcant bits.

II) Loop. Let r run from 0 to pp − 1 and execute steps A and B for each r.

A. Writing. Build up the hash table, which is meant to encode the set Lr, as follows. a) Find all pairs (z,w) of non-negative integers less than or equal to B that satisfy 4 4 z +4w ≡ r (mod pp) and all the congruence conditions for primitive solutions, listed above. (Make systematic use of the Chinese remainder theorem.) b) Execute steps i) and ii) below for each such pair. 4 4 64 i) Evaluate fS(z,w):=(z +4w mod 2 ). ii) Use the hash value h(fS(z,w)) and linear probing to ﬁnd a free place in the hash table, and store the control value c(fS(z,w)) there. B. Reading. Search within the hash table, as follows. a) Find all pairs (x, y) of non-negative integers less than or equal to B that satisfy 4 4 x +2y ≡ r (mod pp) and all the congruence conditions for primitive solutions, listed above. (Make systematic use of the Chinese remainder theorem.) b) Execute steps i) and ii) below for each such pair. 4 4 64 i) Evaluate gS(x, y):=(x +2y mod 2 ) on all points found in step a). ii) Search for the control value c(gS(x, y)) in the hash table, starting at the hash value h(gS(x, y)) and using linear probing, until a free position is found. Report all hits and the corresponding values of x and y.

7.3. Remarks (Some details of the implementation). i) The fourth powers and fourth roots modulo pp are computed during the initialization part of the program and are stored into arrays because arithmetic modulo pp is slower than memory access. ii) The control value is limited to 31 bits as it is implemented as a signed integer. We use the value (−1) as a marker for an unoccupied place in the hash table. iii) In contrast to our previous programs, we do not precompute large tables of fourth powers modulo 264 because access to these tables is slower than the execution of two multiplications in a row (at least on our computer).

178 the diophantine equation x4 +2y4 = z4 +4w4 [Chap. V iv) It is the impact of the congruences modulo 625, 8,and81, described above, that the set of pairs (y, w)[(x, y)] to be read is signiﬁcantly bigger than the set of pairs (x, z)[(z,w)] to be written. They diﬀer actually by a factor of 6252 · 243 · ≈ 6252 · 112 ≈ 2 000·25 112 2 33.901 in case N and 2 000·25 243 3.601 in case S. As a consequence of this, only a small part of the running-time is spent on writing. The lion’s share is spent on unsuccessful searches within L.

7.4. Remarks (Post-processing). i) Most of the hits found in the hash table actually do not correspond to solutions of the Diophantine equation. Hits indicate only a similarity of bit-patterns. Thus, for each pair of x and y reported, one needs to check whether a suitable pair of z and w does exist. We do this by recomputing 4 4 z +4w for all z and w that fulﬁll the given congruence conditions modulo pp and powers of the small primes. Although this method is entirely primitive, only about 3% of the total running-time is actually spent on post-processing. One reason for this is that post-processing is not called very often, on average only once on about ﬁve pages. For those pages, the writing part of the algorithm needs to be recapitulated. This is, however, not time-critical as only a small part of the running-time is spent on writing, anyway. ii) An interesting alternative for post-processing would be to apply the theory of binary quadratic forms. The obvious strategy is to factorize x4 +2y4 completely into prime powers and to deduce from the decomposition all pairs (a, b) such that 2 2 4 4 b a + b = x +2y . Then one may check whether for one of them both a and 2 are perfect squares.

7.5. Remark. The migration to a more bit-based implementation led to an increase of the speed of our programs by a factor of approximately 1.35. ii. Adaptation to the memory architecture of our computer.

7.6. The factor of 1.35 is less than what we actually hoped for. For that reason, we made various tests in order to ﬁnd out what the limiting bottleneck of our program is. It turned out that the major slow-down is the access of the processor to main memory. Our programs are, in fact, doing only two things, integer arithmetic and memory access. The integer execution units of modern processors are highly optimized circuits, and several of them work in parallel inside one processor. They work a lot faster than main memory does. In order to reach a further improvement, it will therefore be necessary to take the architecture of memory into closer consideration.

7.7. The situation. Computer designers try to bridge the gap between the fast processor and the slow memory by building a memory hierarchy, which consists of several cache levels. The cache is a very small and fast memory inside the processor. The ﬁrst cache level, called L1 cache, of our processor consists of a data cache and an instruction cache. Both are 64 kbyte in size. The cache manager stores the most recently used data into the cache in order to make sure a second access to them will be fast.

Sec. 7] improvements ii—adaptation to our hardware 179

If the cache manager does not ﬁnd necessary data within the L1 cache, then the processor is forced to wait. In order to deliver data, the cache management ﬁrst checks the L2 cache, which is 1024 kbyte large. It consists of 16 384 lines of 64 bytes each.

7.8. Our program. Our program fits into the instruction cache, completely. Therefore, no problem should arise from this. When we consider the data cache, however, the situation is entirely different. The cache manager stores the 1024 most recently used memory lines, each being 64 bytes long, within the L1 data cache. This strategy is definitely good for many applications. It guarantees main memory may be scanned at a high speed. On the other hand, for our application, it fails completely. The reason is that access to our 500 Mbyte hash table is completely random. An access directly to the L1 cache happens in by far less than 0.1% of the cases. In all other cases, the processor has to wait. Even worse, it is clear that in most cases we do not even access the L2 cache. This means the cache manager needs to access main memory in order to transfer the corresponding memory line of 64 bytes into the L1 cache. After this, the processor may use the data. In the case that there is no free line available within the L1 cache, the cache manager must restore old data back to main memory, first. This process takes us 60 nanoseconds, at least, which seems to be short, but the processor could execute more than 100 integer instructions during the same time. The philosophy for further optimization must, therefore, be to adapt the programs as much as possible to our hardware, first of all to the sizes of the L1 and L2 caches.

7.9. Programmer’s position. Unfortunately, the whole memory hierarchy is invisible from the point of view of a higher programming language, such as C,since such languages are designed for being machine-independent. Further, the hardware executes the cache management in an automatic manner. This means, even by programming in assembly, one cannot control the cache completely although some new assembly instructions, such as prefetch, allow certain direct manipulations.

7.10. A way out. A practical way, nonetheless to gain some influence on the memory hierarchy, is to rearrange the algorithm in an apparently nonsensical manner, thereby making memory access less chaotic. One may then hope that the automatic management of the cache, when confronted with the modified algorithm, is able to react more properly. This should allow the program to run faster. iii. Our first trial.

7.11. Our ﬁrst idea for this was to work with two arrays instead of one. Algorithm M. i) Store the values of f into an array and the values of g into a another one. Write successively calculated values into successive positions. It is clear that this part of the algorithm is not troublesome as it involves a linear memory access, which is perfectly supported by the memory management.

180 the diophantine equation x4 +2y4 = z4 +4w4 [Chap. V ii) Then use Quicksort in order to sort both arrays. In addition to being fast, Quicksort is known to have a good memory locality when large arrays are sorted. iii) In a ﬁnal step, search for matches by going linearly through both arrays as in Mergesort.

7.12. Remark Unfortunately, the idea behind Algorithm M is too simple to give it any chance of being superior to the previous algorithms. However, it is a worthwhile experiment. Indeed, our implementation of Algorithm M causes at least 30 times more memory transfer compared with the previous programs but, actually, it is only three times slower. This indicates that our approach is reasonable. iv. Hashing with partial presorting.

7.13. Our ﬁnal algorithm is a combination of sorting and hashing. An important aspect of it is that the sorting step has to be considerably faster than the Quicksort algorithm. For that reason, we adopted some ideas from linear-time sorting algorithms, such as the Radix sort or Bucket sort.

7.14. The algorithm works as follows. Again, the description is based on case S, case N being analogous. Algorithm H64B. I. Initialization. Fix B := 108. Initialize a hash table H of 227 = 134 217 728 integers, each being 32 bits long. Fix the page prime pp := 200 003.

In addition, initialize 1024 auxiliary arrays Ai, each of which may contain 217 = 131 072 long (64-bit) integers. Further, define two functions, the hash function h and the control function c,which map 64-bit integers to 27-bit integers and 31-bit integers, respectively, by selecting certain bits. Do not use any of the bits twice to ensure h and c are independent of each other and do not use the four least significant bits. Finally, let h(10) denote the function mapping 64-bit integers to integers within [0, 1023] that is given by the ten most significant bits of h.Inotherwords, for every x, h(10)(x) is the same as h(x) shiftedtotherightby17bits.

II. Outer Loop. Let r run from 0 to pp − 1 and execute A and B for each r.

A. Writing. Build up the hash table, which is meant to encode the set Lr, as follows. a) Preparation. Find all pairs (z,w) of non-negative integers less than or equal to B 4 4 that satisfy z +4w ≡ r (mod pp) and all the congruence-conditions for primitive solutions, listed above. (Make systematic use of the Chinese remainder theorem.) b) Inner Loop. Execute steps i) – iii) below for each such pair. 4 4 64 i) Evaluate fS(z,w):=(z +4w mod 2 ). ii) Do not store fS (z,w) into the hash table immediately. Put

(10) i := h (fS(z,w)),

ﬁrst.

Sec. 7] improvements ii—adaptation to our hardware 181 iii) Add fS(z,w) to the auxiliary array Ai. Maintain Ai as an unordered list. I.e., always write to the lowest unoccupied address.

If there is no space left in Ai, then output an error message and abort the algorithm. c) Storing. Let i run from 0 to 1023.Foreachi let j run through the addresses occupied in Ai.

For ﬁxed i and j, extract from the 64-bit integer Ai[j] the 27-bit hash value h(Ai[j]) and the 31-bit control value c(Ai[j]).

Use the hash value h(Ai[j]) and linear probing to ﬁnd a free place in the hash table and store the control value c(Ai[j]) there. d) Clearing up. Clear the auxiliary arrays Ai for all i ∈ [0, 1023] to make them available for reuse. B. Reading. Search within the hash table, as follows. a) Preparation. Find all pairs (x, y) of non-negative integers less than or equal to B 4 4 that satisfy x +2y ≡ r (mod pp) and all the congruence conditions for primitive solutions, listed above. (Make systematic use of the Chinese remainder theorem.) b) Inner Loop. Execute steps i) – iii) below for each such pair. 4 4 64 i) Evaluate gS(x, y):=(x +2y mod 2 ). ii) Do not look up gS (x, y) in the hash table immediately. Put

(10) i := h (gS(x, y)),

ﬁrst. iii) Add gS (x, y) to the auxiliary array Ai. Maintain Ai as an unordered list. I.e., always write to the lowest unoccupied address.

If there is no space left in Ai, then call d[i] and add gS(x, y) to Ai afterwards. c) Searching. Clearing all buﬀers. Let i run from 0 to 1023.Foreachi, call d[i]. When this is ﬁnished, go to the next iteration of the outer loop.

Subroutine d[i]) Clearing a buﬀer. Let j run through the addresses occupied in Ai. For ﬁxed j, search for the control value c(Ai[j]) within the hash table H,starting at the hash value h(Ai[j]) and using linear probing, until a free place is found. Report all hits and the corresponding values of x and y.

Having done this, declare Ai to be empty.

7.15. Remark The auxiliary arrays Ai play the role of a buﬀer. Thus, one could say that we introduced some buﬀering into the management of the hash table H. However, this description misses the point.

What is more important is that the values of fS to be stored into Lr are partially sorted according to the ten most signiﬁcant bits of h(fS(z,w)) by putting them into the auxiliary arrays Ai. When the hash table is then built up, the records arrive almost in order. The same is true for reading. What we actually did is, therefore, to introduce some partial presorting into the management of the hash table.

182 the diophantine equation x4 +2y4 = z4 +4w4 [Chap. V

7.16. Remark It is our experience that each auxiliary array carries more or less the same load. In particular, in step II.A.b.iii) above, when the buffers are filled up for writing, a buffer overflow should never occur. For this reason, we feel free to treat this possibility as a fatal error. v. Running-time.

7.17. Algorithm H64B uses about three times more memory than our previous algorithms, but our implementation runs almost three times as fast. It was this factor that made it possible to attack the bound B =108 in a reasonable amount of time. The ﬁnal version of our programs took almost exactly 100 days of CPU time on an AMD Opteron 248 processor. This time is composed almost equally of 50 days for case N and 50 days for case S. The main computation was executed in parallel on two machines in February and March 2005.

7.18. Why is this algorithm faster? To answer this question, one has to look at the impact of the cache. For the old program, the cache memory was mostly useless. For the new program, the situation is completely different. When the auxiliary arrays are filled in step II.A.b.iii) and II.B.b.iii), access to these arrays is linear. There are only 1024 of them, which is exactly the number of lines in the L1 cache. When an access does not hit into that innermost cache, then the corresponding memory line is moved to it and the next seven accesses to the same auxiliary array are accesses to that line. Altogether, seven of eight memory accesses hit into the L1 cache. When an auxiliary array is emptied in step II.A.c) or II.B.d[i]), the situation is similar. There are a high number of accesses to a very short segment of the hash table. This segment fits completely into the L2 cache. It has to be moved into that cache once. Then it can be used many times. Again, access to the auxiliary array is linear and a hit into the L1 cache occurs in seven of eight cases. All in all, for Algorithm H64B, most memory accesses are hits into the cache. This means, at the cost of some more data transfer altogether, we achieved that main memory may be mostly used at the speed of the cache.

8. The solution found

8.1. Ironically, the premature version of our algorithm as described in Sec- tion 4 already solved Problem 1.4. The improvements made it possible to shift the search bound to 108,whichwasfar beyond our original expectations. However, they did not produce any new solutions. Thus, let us conclude this chapter by some comments on the simpler approach.

Sec. 8] the solution found 183

8.2. Test versions of the program were written in Delphi. The deﬁnitive version was written in C. It took about 130 hours of CPU time on a 3.00 GHz Pentium 4 processor with 512-kbyte cache memory. The main computation was executed in parallel on two machines during the very ﬁrst days of December 2004.

8.3. Instead of looking for solutions of x4 +2y4 = z4 +4w4, the algorithm searches, in fact, for solutions to the corresponding simultaneous con- 4 4 gruences modulo pp and pc, which, in addition, fulﬁll that (x +2y ) mod ph 4 4 and (z +4w ) mod ph are “almost equal”. To this modiﬁed problem, we found approximately 3800 solutions such that (y, w) =(0 , 0). These congruence solutions were checked by an exact computation using O. Forster’s [For] Pascal-style multiprecision interpreter language ARIBAS.

8.4. Among the congruence solutions, exact equality occurred only once. This solution is as follows. ==> 1484801**4 + 2 * 1203120**4. -: 90509_10498_47564_80468_99201

==> 1169407**4 + 4 * 1157520**4. -: 90509_10498_47564_80468_99201

CHAPTER VI

Points of bounded height on cubic and quartic threefolds∗

... , one by one, or all at once. W. S. Gilbert and A. Sullivan: The Yeomen of the Guard (1888)

1. Introduction—Manin’s conjecture i. Summary.

1.1. For the families ax3 = by3 + z3 + v3 + w3, a, b =1, ... ,100,and ax4 = by4 + z4 + v4 + w4, a, b =1, ... ,100, of projective algebraic threefolds, we test numerically the conjecture of Manin, in the reﬁned form due to E. Peyre, cf. Conjecture II.7.3, about the asymptotics of points of bounded height on Fano varieties. This includes searching for points, computing the Tamagawa number, and detecting the accumulating subvarieties. The goal of this chapter is describe these computations as well some background on the geometry of cubic and quartic threefolds. ii. Manin’s conjecture.

1.2. We discussed Manin’s conjecture in full generality in Chapter II. The particular situation of the present chapter is, however, signiﬁcantly easier than the general case. For this reason, we feel that an independent introduction, designed for this speciﬁc case, should be helpful. In an ideal case, we hope that it makes this chapter readable independently of Chapter II.

1.3. Consider a projective algebraic variety X over É, and ﬁx an embedding

ι: X → Pn . Recall that, in this situation, there is the well-known naive height

É → Ê Hnaive : X( É) given by

Hnaive(P ):= max | xi| . i=0,... ,n n Here, (x0 : ... : xn):=ι(P ) ∈ P ( É), where the projective coordinates are integers satisfying gcd(x0,...,xn)=1.

∗This chapter is a revised and slightly extended version of the article, “The asymptotics of points of bounded height on diagonal cubic and quartic threefolds”, in Algorithmic number theory, Lecture Notes in Computer Science 4076, Springer, Berlin 2006, 317–332, with A.-S. Elsenhans and the author.

185

186 points of bounded height on threefolds [Chap. VI

It is of interest to ask for the asymptotics of the number of É-rational points on X of bounded naive height. This applies particularly to Fano varieties as those are expected to have many rational points, at least after a ﬁnite extension of the ground- ﬁeld.

1.4. Simplest examples of Fano varieties are complete intersections in Pn of É a multidegree (d1,...,dr) such that d1 + ...+ dr ≤ n. In this case, the conjecture of Manin reads as follows.

1.5. Conjecture. Let X ⊆ Pn be a non-singular complete intersection of É multidegree (d1,...,dr). Assume dim X ≥ 3 and k := n +1− d1 − ... − dr > 0. Then there exists a Zariski open subset X◦ ⊆ X such that

{ ∈ ◦ | k }∼ ∗ # x X ( É) Hnaive(x)

1.6. Example. Let X ⊂ P4 be a smooth hypersurface of degree 4. É Conjecture 1.5 predicts ∼ τB rational points of height

1.7. Remark. Conjecture 1.5 is proven in a number of particular situations. Classical cases are the projective space itself, linear subspaces, quadrics, and complete intersections of a multidegree (d1,...,dr), where the dimension of X is very large compared to d1,...,dr [Bir]. The general version of the conjecture (Conjecture II.7.3) is known to be true in a number of further particular cases. A rather complete description of the knowledge of the year 2002 may be found in the survey article [Pe02, Sec. 4] of E. Peyre. But, in more recent time, a lot of further progress has been made and the whole ﬁeld seems to be in rapid development. We brieﬂy reported on the typical strategies for proving Manin’s conjecture in Section II.8. Nevertheless, Conjecture 1.5 is still unknown, for example, in the situation of a general cubic surface. In this chapter, we present numerical evidence for Conjecture 1.5 in the case of the varieties Xe ,givenbyaxe = bye + ze + ve + we in P4 for e =3and 4. a,b É

1.8. Remark. By the Noether–Lefschetz Theorem, the assumptions made ∼

on X imply that Pic(X ) = [Ha70, Corollary IV.3.2]. This is no longer true in É dimension two. See Remark 1.15.ii) for more details.

Sec. 1] introduction—manin’s conjecture 187 iii. The constant.

1.9. Conjecture 1.5 is compatible with results obtained by the classical circle method (e.g., [Bir]). Motivated by this, E. Peyre provided a description of the constant τ expected in (∗). We formulated the deﬁnition for the general case in Deﬁnition II.7.1. ∼

In the situation considered here, Pic(X ) = implies that β(X)=1and there É is no Brauer–Manin obstruction on X. Peyre’s constant is therefore equal to the Tamagawa-type number

· 1 − É τ(X):=α(X) (1 p ) ωp(X( p)) . ∪{∞} p∈È O − ∼ In this formula, α(X):=1/k for k ∈ Æ such that ( K) = O(k). The measure ω is given in local p-adic analytic coordinates x , ... ,x by p 1 d ∂ ∂ ∧ ... ∧ dx1 ... dxd . ∂x1 ∂xd p

Here, each dxi denotes a Haar measure on Ép that is normalized in the usual manner. ∂ ∧ ... ∧ ∂ is a section of O(−K). ∂x1 ∂xd

1.10. For p finite, one has a natural model X ⊆ Pn of X,givenbythe p defining equation. This induces the metric . p on O(k). It is almost immediate from the definition that

m #X ( /p )

ωp X( Ép) = lim . m→∞ pm dim X

1.11. Remark. There is another description of ωp of interest for ﬁnite p. One has − −k

1 p m (d+1)m

X ωp X( Ép) = lim #C ( /p )/p . 1 − p−1 m→∞ Here, CX denotes the aﬃne cone over X . For a proof, see [Pe/T, Corollary 3.5].

1.12. The hermitian metric . ∞ corresponding the naive height Hnaive is k O − ∼ O given by . ∞ := . min on ( K) = (k). Here, . min is the hermitian metric on O(1) deﬁned by xi min := inf | xi/xj | . j=0,... ,n

The factor ω∞ X( É∞) may then be described as follows.

1.13. Lemma. If X ⊂ Pn is a hypersurface deﬁned by the equation f =0, then · 1 α(X) ω∞ X( Ê) = ω . 2 Leray f(x0,... ,xn)=0, | x0|,... ,| xn|≤1

The Leray measure ωLeray on

{ n+1 | } (x0,...,xn) ∈ Ê f(x0, ... ,xn)=0

188 points of bounded height on threefolds [Chap. VI

1 is related to the usual hypersurface measure by the formula ωLeray = grad f ωhyp. On the other hand, one may also write

1 ∧ ∧ ∧ ∧ ωLeray = ∂f dx0 ... dxi ... dxn . | (x0,...,xn)| ∂xi

Proof. The equivalence of the two descriptions of the Leray measure is shown in Lemma II.6.15. The main assertion is proven in Proposition II.6.19.

1.14. Remark. As the description of the constant τ given here grew out of a deﬁnition given in canonical terms, it is no surprise that τ is invariant under scaling.

This can also be seen directly. When we multiply f by a prime number p,thenτp gets multiplied by a factor of p. On the other hand, τ∞ gets multiplied by a factor of 1/p and all the other factors remain unchanged.

1.15. Remark. There are several ways to generalize Conjecture 1.5. These are presented in detail in Chapter II. For completeness, let us recall the following facts. i) One may consider more general heights corresponding to the tautological invertible sheaf O(1). This includes a) replacing the minimum metric by an arbitrary continuous hermitian metric on O(1). This would affect the domain of integration for the factor at infinity. b) multiplying Hnaive(x) with a function that depends on the reduction of x modulo ∈ some N Æ. This augments Conjecture 1.5 by an equidistribution statement. ii) Instead of complete intersections, one may consider arbitrary projective Fano va- k rieties X.ThenHnaive needs to be replaced by a height defined by the anticanonical sheaf O(−KX ). ∼

If Pic(X ) = , then the description of the constant C gets more complicated in É

1 É

several ways. First, there is an additional factor β := #H Gal( É/ ), Pic(X ) . É − 1 Further, instead of (1 p ), one has to use more complicated convergence-gener-

ating factors. (Cf. Remark II.7.5.i).) Finally, the Tamagawa measure has to be

taken not of the full variety X( É ) but of the subset that is not aﬀected by the Brauer–Manin obstruction.

∼

É ∗ If Pic(X) = already over , then the right-hand side of ( ) has to be replaced by CB logt B. For the exponent of the log-term, there is the expectation that t = rk Pic(X) − 1. There are, however, examples [Ba/T96] in dimension three, in which the exponent is larger.

The definition of α is rather complicated, in general. The factor α depends on the structure of the effective cone in Pic(X) and on the position of (−KX ) in it. (Cf. Definition II.5.2).

Sec. 2] computing the tamagawa number 189

2. Computing the Tamagawa number i. Counting points over ﬁnite ﬁelds.

e 2.1. We consider the projective varieties Xa,b given by

axe = bye + ze + ve + we in P4 . We assume a, b =0 (and p e) in order to avoid singularities. Observe that p 2 even for large p these are at most e varieties up to obvious p-isomorphism as

∗ ∗ e p consists of no more than e cosets modulo ( p ) . It follows from the Weil conjectures, proven by P. Deligne [Del, Théorème (8.1)], that e 3 2 e #Xa,b( p)=p + p + p +1+Ea,b

e | e |≤ 3/2 where the error-term Ea,b may be estimated by Ea,b Cep . 3 3 Here, C3 =10and C4 =60as dim H (X , Ê)=10for every smooth cubic

3 4 4 threefold and dim H (X , Ê)=60for every smooth quartic threefold in P ( ). These dimensions result from the Weak Lefschetz Theorem together with F. Hirze- bruch’s formula [Hi, Satz 2.4] for the Euler characteristic, which actually works in much more generality.

2.2. Remark. Suppose e =3and p ≡ 2(mod 3).Then

3 1 #Xa,b( p)=#Xa,b( p) as gcd(p−1, 3) = 1. Similarly, for e =4and p ≡ 3(mod 4), one has gcd(p−1, 4) = 2 and

4 2 #Xa,b( p)=#Xa,b( p).

In these cases, the error term vanishes and

e 3 2 #Xa,b( p)=p + p + p +1.

In the remaining cases p ≡ 1(mod 3) for e =3and p ≡ 1(mod 4) for e =4,our e e ⊆ 4 goal is to compute the number of p-rational points on Xa,b.AsXa,b P ,there would be an obvious O(p4)-algorithm. We can do signiﬁcantly better than that.

2.3. Definition. Let K be a field, and let x ∈ Kn and y ∈ Km be ∗ ∈ n+m−1 two vectors. Then their convolution z := x y K is defined to be zk := i+j=kxiyj .

2.4. Theorem (FFT convolution). Let n =2l,andletK be a ﬁeld containing the 2n-th roots of unity. Then the convolution x ∗ y of two vectors x, y of length ≤ n can be computed in O(n log n) steps. Proof. The idea is to apply the Fast Fourier Transform (FFT) [For, Satz 20.3]. The connection to the convolution is shown in [For, Satz 20.2] or [C/L/R,Theo- rem 32.8].

190 points of bounded height on threefolds [Chap. VI

Theorem 2.4 is the basis for the following algorithm.

e 2.5. Algorithm (FFT point counting on Xa,b). i) Initialize a vector x[0 ...p] with zeroes. ii) Let r run from 0 to p − 1, and increase x[re mod p] by 1. iii) Calculate y˜ := x ∗ x ∗ x by FFT convolution. iv) Normalize by putting y[i]:=˜y[i]+˜y[i + p]+˜y[i +2p] for each i ∈{0,...,p− 1}. v) Initialize N as zero. vi) (Counting points with first coordinate =0 )Letj run from 0 to p − 1,and increase N by y[(a − bj4) mod p]. vii) (Counting points with first coordinate 0 and second coordinate =0 ) Increase N by y[(−b) mod p]. viii) (Counting points with first and second coordinate 0) Increase N by (y[0] − 1)/(p − 1). e ix) Return N as the number of all p-valued points on Xa,b.

2.6. Remark. For the running-time, step iii) is dominant. Therefore, the running-time of Algorithm 2.5 is O(p log p). e To count, for ﬁxed e and p, p-rational points on Xa,b with varying a and b, one needs to execute the ﬁrst four steps only once. Afterwards, one may perform steps v) through ix) for all pairs (a, b) of elements from a system of representatives

∗ ∗ e for p /( p ) . Note that steps v) through ix) alone are of complexity O(p).

2.7. We ran this algorithm for all primes p ≤ 106 and stored the cardinalities in a ﬁle.

e 2.8. Remark. There is a formula for #Xa,b( p) in terms of Jacobi sums. A skillful manipulation of these sums should lead to another eﬃcient algorithm, which serves the same purpose as Algorithm 2.5. ii. The local factors at ﬁnite places.

2.9. We are interested in the Euler product

e m #X ( /p ) e − 1 a,b τa,b,ﬁn := 1 lim . p m→∞ p3m

p∈È

2.10. Lemma. a) (Good reduction)

X e m 3m If p abe, then the sequence (# a,b( /p )/p )m∈Æ is constant.

Sec. 2] computing the tamagawa number 191 b) (Bad reduction)

X e m 3m i) If p divides ab but not e, then the sequence (# a,b( /p )/p )m∈Æ becomes stationary as soon as pm divides neither a nor b.

X e m 3m ii) If p =2and e =4, then the sequence (# a,b( /p )/p )m∈Æ becomes stationary as soon as 2m does not divide 8a or 8b.

X e m 3m iii) If p =3and e =3, then the sequence (# a,b( /p )/p )m∈Æ becomes stationary as soon as 3m divides neither 3a nor 3b. iii. An estimate.

2.11. Theorem. For every pair (a, b) of integers such that a, b =0 ,the e Euler product τa,b,ﬁn is convergent. Proof. Cf. Lemma II.6.23 where this is proven in more generality. Let p be a prime bigger than | a|, | b|,ande. Then the factor at p is

− 1 2 3 3/2 3 τp := (1 p )(1 + p + p + p + Dpp )/p , where |Dp|≤Ce for C3 =10and C4 =60, respectively. As sums are easier to estimate than products, we take a look at the logarithm,

D log τ = p + O(p−5/2) . p p3/2 Taking the logarithm, we consider p log τp.Inthecasee =3, the sum is eﬀec- tively over the primes p =1(mod 3).Ife =4, then summation extends over all primes p =1(mod 4). In either case, we take a sum over one-half of all primes. This leads to the estimate 4 5 ∞ C C 1 | log τ |≤ e + O(p−5/2) ∼ e dt p 3/2 3/2 ≥ ≥ p 2 t log t p N p N N ∞ C C ≤ e t−3/2dt = √ e . 2logN N log N N

2.12. Remark. We are interested in an explicit upper bound for log τp . p≥106 Using Taylor’s formula, one gets D log τ − p ≤ 10−8. p p3/2 p≥106 p≥106 Dp ≡ ≡ Since p3/2 is zero for p 3(mod 4) (or p 2(mod 3)), the sum should be compared with log(ζK (3/2)). Here, ζK is the Dedekind zeta function of K = É(i) or

192 points of bounded height on threefolds [Chap. VI

É(ζ ), respectively. This yields 3

1 ζ É (3/2) ≤ log (i) p3/2 (1 − 2−3/2)−1/2 · (1 − p−3)−1/2 · (1 − p−3/2)−1 p≥106 p≡3(mod 4) p<106 p≡1(mod 4) p≡1(mod 4)

and, for the other case, a similar estimate containing ζ É(ζ3)(3/2). Note that the inﬁnite product in the denominator converges a lot faster than the left-hand side. Using Pari, we evaluated the right-hand side numerically. We found 0.39% for the quartic and 0.065% for the cubic as upper bounds for the error of approximation.

2.13. Remark. In practice, the error of the approximation is much smaller. The main reason is that the error-term Dp may have a positive or a negative sign. Some cancellations happen during summation. The assumption of a random distribution would result in a higher order of convergence. In fact, we observed this eﬀect numerically. iv. Approximation of the Euler product.

2.14. Lemma 2.10 allows us to determine each factor of the Euler prod- e uct exactly. As we need to know the numerical value of τa,b,ﬁn, we approximate it by a ﬁnite product. − · e 3 Observe that the factor at a good prime p is simply (1 1/p) #Xa,b( p)/p . In particular, for this factor there are only e2 values possible. Even more, these numbers had been precomputed using FFT point counting (Algorithm 2.5). The algorithm below is based on the fact that the vast majority of the factors actually do not need to be computed. They are available from a list.

3 4 2.15. Algorithm (Compute an approximation for τa,b,ﬁn (τa,b,ﬁn)). i) Let p run over all prime numbers such that p ≡ 2(mod 3) (p ≡ 3(mod 4))andp ≤ N, and calculate the product of all values of (1 − 1/p4). ii) Compute the factor corresponding to p =3(p =2) by Lemma 2.10.b). iii) Let p run over all prime numbers such that p ≡ 1(mod 3) (p ≡ 1(mod 4)) and p ≤ N. Calculate the product of the factors described below. If p|ab, then the corresponding factor is given by Lemma 2.10.b). Otherwise, compute the e-th power residue symbols of a and b and look up the precomputed factor

for this p-isomorphism class of varieties in the list. iv) Multiply the two products from steps i) and iii) and the factor from step ii) with each other. Correct the product by taking the bad primes p ≡ 2(mod 3) (p ≡ 3(mod 4)) into consideration.

m 2.16. Remark. When we meet a bad prime p,wehavetocount/p - X e valued points on a,b. This is done by an algorithm that is very similar to Algo- rithm 2.5.

3 2.17. We used Algorithm 2.15 to compute the Euler products τa,b,ﬁn and 4 6 τa,b,ﬁn for a, b =1, ... ,100. We did all calculations for N =10 . Note that step i) had to be done only once for e =3and once for e =4.

Sec. 3] on the geometry of diagonal cubic threefolds 193 v. The factor at the inﬁnite place.

2.18. In order to get an approximation for the integrals, we tried to use some standard methods from numerical analysis [Kr, Chapter 9], particularly the Gauß–Legendre formula. A direct application of this method is known to work well as long as the integrand is smooth enough and the dimension of the domain of integration is not too big.

2.19. For the quartic X4 , we have the integral a,b 4 √1 1 ω∞ X ( Ê) = dy dz dv dw a,b 4 4 a (by4 + z4 + v4 + w4)3/4 R over

4 || | | | | | | |≤ | 4 4 4 4|≤ } R := {(y, z, v, w) ∈ Ê y , z , v , w 1 and by + z + v + w a .

The integrand is singular in one point. We used a simple substitution to make it suﬃciently smooth for numerical integration.

On the other hand, for the cubic X3 , we have to consider a,b 3 √1 1 ω∞ X ( Ê) = dy dz dv dw a,b 6 3 a (by3 + z3 + v3 + w3)2/3 R { ∈ 4 ||| | | | | | |≤ | 3 3 3 3|≤ } for R := (y, z, v, w) Ê y , z , v , w 1 and by + z + v + w a . The diﬃculty here is the handling of the singularity of the integrand. It is located in the zero set of by3 + z3 + v3 + w3 in R, which is a cone over a cubic surface. Since (by3 + z3 + v3 + w3)−2/3 is a homogeneous function, it is enough to integrate over the boundary of R. This reduces the problem to several three-dimensional integrals of functions having a two-dimensional singular locus. If a ≥ b +3,thenR is a cube and the boundary of R is easy to describe. We restricted our attention to this case. We smoothed the singularities by separation of Puiseux expansions and substitutions. The resulting integrals were treated by the Gauß–Legendre formula [Kr].

3. On the geometry of diagonal cubic threefolds i. Surfaces on a hypersurface in P4.

3.1. Lemma. Let X ⊂ P4 be any smooth hypersurface. Then every (reduced but possibly singular) surface S ⊂ X is a complete intersection X ∩ Hd with a 4 hypersurface Hd ⊂ P . ∼

Proof. By the Noether–Lefschetz Theorem, we have Pic(X) = . The surface S is a Weil divisor on X. Hence, O(S)=O(d) ∈ Pic(X) for a certain d>0.There- 4 1 4 striction Γ(P , O(d)) → Γ(X, O(d)) is surjective as H (P , OX (d − deg X)) = 0 [Ha77, Theorem III.5.1.b)].

194 points of bounded height on threefolds [Chap. VI ii. Elliptic cones.

3.2. Let X ⊂ P ( ) be the diagonal cubic threefold given by the equation 3 3 3 3 3 3 x + y + z + v + w =0.Fixζ ∈ such that ζ =1. Then, for every 3 3 3 point (x0 : y0 : z0) on the elliptic curve F : x + y + z =0, the line given by (x : y : z)=(x0 : y0 : z0) and v = −ζw is contained in X. All these lines together form a cone CF over F , the cusp of which is (0 : 0 : 0 : −ζ :1). CF is a singular model of a ruled surface over an elliptic curve. This shows there are no other rational curves contained in CF. By permuting coordinates, one ﬁnds a total of thirty elliptic cones of that type within X. The cusps of these cones are usually named Eckardt points [Mu-e, Cl/G]. We call the lines contained in one of these cones the obvious lines lying on X. It is clear that there are an inﬁnite number of lines on X running through each of the thirty Eckardt points (1:−1:0:0:0), (1:0:−1:0:0), ...,(0:0:0:1:−1), (1:−e2πi/3 :0:0:0), ...,(0:0:0:1:−e−2πi/3).

3.3. Proposition (cf. [Mu-e, Lemma 1.18]). Let X ⊂ P4 be the diagonal cubic threefold given by the equation x3 + y3 + z3 + v3 + w3 =0. Then through each point P ∈ X diﬀerent from the thirty Eckardt points there are precisely six lines on X.

Proof. Let P =(x0 : y0 : z0 : v0 : w0). A line l through P and another point Q =(x : y : z : v : w) is parametrized by (s : t) → ((sx0 + tx): ... :(sw0 + tw)). Comparing coeﬃcients at s2t, st2,andt3, we see that the condition that l lies on X maybeexpressedbythethreeequationsbelow.

2 2 2 2 2 † x0x + y0y +z0 z + v0v + w0w =0, ( ) 2 2 2 2 2 x0x + y0y + z0z + v0v + w0w =0, (‡) x3 + y3 + z3 + v3 + w3 =0. (§)

The ﬁrst equation means that Q lies on the tangent hyperplane HP at P , while equation (§)justencodesthatQ ∈ X. By Lemma 3.6, HP ∩ X is an irreducible cubic surface. On the other hand, the quadratic form q on the left-hand side of equation (‡)isof rank at least 3 as P is not an Eckardt point. Therefore, q is not just the product | ≡ of two linear forms. In particular, q HP 0. ∩ | ∩ As HP X is irreducible, Z(q HP ) and HP X do not have a component in common. By Bezout’s theorem, their intersection in HP is a curve of degree 6.

3.4. Remark. It may happen that some of the six lines coincide. Actually, it turns out that a line appears with multiplicity > 1 if and only if it is obvious [Mu-e, Lemma 1.19]. In particular, for a general point P the six lines through it are diﬀerent from each other. Under certain exceptional circumstances√ it is possible to write down all six 3 lines√ explicitly. For example, if P =( −4:1:1:1:1), then the line ( 3 −4t :(t + s):(t + is):(t − s):(t − is)) through P lies on X. Permuting the three rightmost coordinates yields all six lines.

Sec. 4] accumulating subvarieties 195

3.5. Remark. The following lemma is a special case of Zak’s theorem [Za, Corollary 1.8], which is more elementary and completely suﬃcient for our purposes. We present it here for the convenience of the reader.

3.6. Lemma. Let X ⊂ P4 be the diagonal cubic threefold given by the equation x3 +y3 +z3 +v3 +w3 =0. Then, for any hyperplane H ⊂ P4, the intersection H ∩ X is irreducible and has at most ﬁnitely many singular points. Proof. The intersection of X with a hyperplane H is singular precisely in those points where H is tangent to X.

The tangent hyperplane H at (x0 : y0 : z0 : v0 : w0) ∈ X is given by 2 2 2 2 2 x0x + y0y + z0 z + v0v + w0w =0. From this formula, we see that H is tangent to X at all points of the form (±x0 : ±y0 : ±z0 : ±v0 : ±w0) that happen to lie on X and at no others. By consequence, H ∩ X admits only a ﬁnite number of singular points. Every irreducible component of H ∩X is a hypersurface in the projective 3-space H. Two such components would intersect in a curve of singular points. Therefore, H∩X is necessarily irreducible.

3.7. Remark. For a general point p ∈ X, the intersection of its tangent hyperplane with X admits exactly one singular point. Indeed, let p =(x0 :y0 :z0 :v0 :w0).

If (−x0 : y0 : z0 : v0 : w0) ∈ X,thenx0 =0.If(−x0 : −y0 : z0 : v0 : w0) ∈ X,then x0 + ζy0 =0for ζ a third root of unity or ζ =1. At this point, up to permutation of coordinates, the list of all possibilities is already complete. Multitangent hyperplanes are caused only by points lying on a ﬁnite arrangement of hyperplanes.

4. Accumulating subvarieties

i. The detection of É-rational lines on the cubics.

3 4.1. On a cubic threefold Xa,b, quadratic growth is predicted for the number

of É-rational points of bounded height. Lines are the only curves with such a growth rate. The moduli space of the lines on a cubic threefold is well understood. It is a surface of general type [Cl/G, Lemma 10.13]. Nevertheless, we do not know of a method

to ﬁnd all É-rational lines on a given cubic threefold, explicitly. For that reason, we use the algorithm below, which is an irrationality test for the six lines through

3 ∈ É a given point (x0 :y0 :z0 :v0 :w0) Xa,b( ).

4.2. Algorithm (Test the six lines through a given point for irrationality). i) Let p run through the primes from 3 to N.

3 5 † ‡ § For each p, solve the system of equations ( ), ( ), ( ) (adapted to Xa,b)in p .Ifthe multiples of (x0,y0,z0,v0,w0) are the only solutions, then output that there is no

É-rational line through (x0 :y0 :z0 :v0 :w0) and terminate prematurely. ii) If the loop comes to its regular end, then output that the point is suspicious.

It could possibly lie on a É-rational line.

196 points of bounded height on threefolds [Chap. VI

4.3. Remark. We use a very naive O(p)-algorithm to solve the system of equations over p.If,say,x0 =0, then it is suﬃcient to consider quintuples such that x =0. We parametrize the projective plane given by (†). Then we compute all points on the conic given by (†)and(‡). For each such point, we compute the cubic form on the left-hand side of (§). When a non-trivial solution is found, we stop immediately.

We carried out the irrationality test on every É-rational point found on any of the cubics except for the points lying on an obvious line. We worked with N = 600. It turned out that suspicious points are rare and that, at least in our sample, each

of them actually lies on a É-rational line.

The lines found. We found only 42 non-obvious É-rational lines on all of the cubics 3 ≥ ≥ ≥ Xa,b for 100 a b 1 together. Among them, there are only five essentially different ones. We present them in Table 1 below. The list might be enlarged by 3 3 3 3 two, as X21,6 and X22,5 may be transformed into X48,21 and X40,22, respectively, by an automorphism of P4. Further, each line has six pairwise different images under the obvious operation of the group S3.

Table 1. Sporadic lines on the cubic threefolds a b Smallest point Point s.t. x =0 19 18 (1:2:3:-3:-5) (0 : 7 : 1 : -7 : -18) 21 6 (1:2:3:-3:-3) (0 : 9 : 1 : -10 : -15) 22 5 (1:-1: 3: 3: -3) (0 : 27 : -4 : -60 : 49) 45 18 (1:1:3:3:-3) (0 : 3 : -1 : 3 : -8) 73 17 (1 : 5 : -2 : 11 : -15) (0 : 27 : -40 : 85 : -96)

4.4. Remark. It is a priori unnecessary to search for accumulating surfaces, at least if we assume the conjectures of Batyrev/Manin and Lang, formulated in Chapter II. First of all, only rational surfaces are supposed to accumulate that many rational points that it could be seen through our asymptotics of O(B2). Indeed, a surface that is abelian or bielliptic may not have more than O(logtB) points of height

Sec. 4] accumulating subvarieties 197

Therefore, if d ≥ 3,then

d/3 d/3 Hnaive(ϕ(p)) = HdH−E(p)=H − 3 (p) ≥ c · H−K (p) 3H d E

t 3 2 for p ∈ supp(E). Manin’s conjecture implies there are O((B log B) d )=o(B ) points of height

ii. The detection of É-rational conics on the quartics.

4.5. On a quartic threefold, linear growth is predicted for the number of

É-rational points of bounded height. The assumption b>0 ensures that there are 4 no Ê-rational lines contained in Xa,b. The only other curves with at least linear growth one could think about are conics.

We do not know of a method to ﬁnd all É-rational conics on a given quartic threefold, explicitly. Worse, we were unable to create an eﬃcient routine to test

whether there is a É-rational conic through a given point. The resulting system of equations seems to be too complicated to handle.

Conics through two points. AconicQ through (x0 : y0 : z0 : v0 : w0) and (x1 :y1 :z1 :v1 :w1) may be parametrized in the form

2 2 2 2 (s : t) → ((λx0s + μx1t + xst): ... :(λw0s + μw1t + wst))

∈ 4 for some x, y, z, v, w, λ, μ . The condition that Q is contained in Xa,b leads to asystemG of seven equations in x, y, z, v, w,andλμ. The phenomenon that λ and μ do not occur individually is explained by the fact that they are not invariant under the automorphisms of P1 ﬁxing 0 and ∞.

4.6. Algorithm (Test for conic through two points). i) Let p run through the primes from 3 to N. In the exceptional case that G could allow a solution such that p|x, y, z, v, w but

2 6 p λμ, do nothing. Otherwise, solve G in p .If(0, 0, 0, 0, 0, 0) is the only solution,

then output that there is no É-rational conic through (x0 : y0 : z0 : v0 : w0) and (x1 :y1 :z1 :v1 :w1), and terminate prematurely. ii) If the loop comes to its regular end, then output that the pair is suspicious.

It could possibly lie on a É-rational conic.

198 points of bounded height on threefolds [Chap. VI

6 4.7. To solve the system G in p ,weuseanO(p)-algorithm. Actually, comparison of coeﬃcients at s7t and st7 yields two linear equations in x, y, z, v,andw. We parametrize the projective plane I given by them. Comparison of coeﬃcients at s6t2 and s2t6 leads to a quadric O and an equation λμ = q(x, y, z, v, w)/M with | a quadratic form q over and an integer M =0.Thecasep M sends us to the next prime immediately. Otherwise, we compute all points on the conic I ∩ O. For each of them, we test the three remaining equations. When a non-trivial solution is found, we stop immediately.

4 Conics through three points. Three points P1, P2,andP3 on Xa,b deﬁne a projec- ∩ ∩ tive plane P. The points together with the two tangent lines P TP1 and P TP2 determine a conic Q, uniquely. It is easy to transform this geometric insight into a formula for a parametrization of Q. We then need to test whether a conic given 4 in parametrized form is contained in Xa,b. This part is algorithmically simple but requires the use of multiprecision integers.

4 Detecting conics. For each quartic Xa,b,wetestedeverypairofÉ-rational points of height < 100 000 for a conic through them. The existence of a conic through (P, Q) is equivalent to the existence of a conic through (gP, gQ) for

∈ 4 4 ⊆ g ( /2 ) S3 Aut(Xa,b). This reduces the running-time by a factor of about 96. Further, pairs already known to lie on the same conic were excluded from the test. For each pair (P, Q) found suspicious, we tested the triples (P, Q, R) for R running

through the É-rational points of height < 100 000 until a conic was found. Due to the symmetries, one ﬁnds several conics at once. For each conic detected, all points on it were marked as lying on this conic. Actually, there were a few pairs found suspicious, through which no conic could be found. In any of these cases, it was easy to prove by hand that there is actually

no Ê-rational conic passing through the two points. This means we detected every conic that meets at least two of the rational points of height < 100 000. Concerning our programming eﬀorts, this was the most complex part of the entire project.

The conics found. Up to symmetry, we found a total of 1 664 É-rational conics 4 ≤ ≤ on all of the quartics Xa,b for 1 a, b 100 together. Among them, 1 538 are contained in a plane of type z = v + w and Yx− Xy =0for (X, Y, t) a rational point on the genus one curve aX4 − bY 4 =2t2. Further, there are 93 conics that are slight modifications of the above with y interchanged with z, v,orw.Thisis possible if b is a fourth power. There is a geometric explanation for the occurrence of these conics. The hyperplane given by z = v + w intersects X4 in a surface S with the two singular points a,b _ _/ (0 : 0 : −1:e±2πi/3 : e∓2πi/3). The linear projection π : S P1 to the first two coordinates is undefined only in these two points. Its fibers are plane quartics

Sec. 5] results 199 splitting into two conics as (v + w)4 + v4 + w4 =2(v2 + vw + w2)2.Afterresolution of singularities, the two conics become disjoint. S0 is a ruled surface over a twofold cover of P1 ramified in the four points such that ax4 − by4 =0. I.e., over a curve of genus one. In the case that a is twice a square, a different sort of conics comes from the equations v = z + Dy and w = Ly when (L, D) is a point on the affine genus three 4 4 4 4 4 4 curve Cb : L + b = D . Here, by + z + v + w becomes twice a perfect square 4 after the substitutions. This explains why this particular intersection of Xa,b with a plane splits into two conics. We found 28 conics of this type. Cb has a É-rational

point for b =5, 15, 34, 39, 65, 80,and84. The conics actually admit a É-rational point for a =2, 18, 32,and98. The remaining ﬁve conics are given as follows. For a =3, 12, 27,or48 and b =10, intersect with the plane given by v = y + z and w =2y + z.Fora =17and b =30, put v =2x + y and w = x +3y + z.

4.8. Remark. Again, it is not necessary to search for accumulating surfaces. _ _/ Here, rational maps ϕ: P2 X ⊂ P4 such that deg ϕ ≤ 3 need to be taken into consideration. We claim, such a map is impossible. If deg ϕ =3,thenwehad

ϕ:(λ:μ:ν) → (K0(λ, μ, ν):... : K4(λ, μ, ν)),

where K0, ... ,K4 are cubic forms defined over É. K0 =0defines a plane cubic, which has infinitely many real points, automatically. As the image of ϕ is assumed 4 to be contained in Xa,b,wehavethatK0(λ, μ, ν)=0implies

K1(λ, μ, ν)= ... = K4(λ, μ, ν)=0for λ, μ, ν ∈ Ê.

By consequence, K1,...,K4 are divisible by K0 (orbyalinearfactorofK0 in the case it is reducible) and ϕ is not of degree three. For deg ϕ ≤ 2,wehaddeg ϕ(P2) ≤ 4 such that ϕ(P2)=X ∩H is a hyperplane section. Lemma 3.6 shows it has at most ﬁnitely many singular points. On the other hand, a quartic in P3 that is the image of a quadratic map from P2 is a Steiner surface. It is known [Ap, p. 40] to have one, two, or (in generic case) three singular lines.

5. Results i. A technology to ﬁnd solutions of Diophantine equations.

5.1. In Chapter V (cf. [EJ2]and[EJ3]), we described a modiﬁcation of D. Bernstein’s [Be] method to search eﬃciently for all solutions of naive height

f(x1,...,xn)=g(y1,...,ym) .

The expected running-time of our algorithm is O(Bmax{n,m}). Its basic idea is as follows.

200 points of bounded height on threefolds [Chap. VI

5.2. Algorithm (Search for solutions of a Diophantine equation). n || | } i) (Writing) Evaluate f on all points of the cube {(x1,...,xn) ∈ xi

e 5.3. Remark. In the case of a variety Xa,b, the running-time is obviously O(B3). We decided to store the values of ze + ve + we into the hash table. Afterwards, we have to look up the values of axe − bye. In this form, the algorithm would lead to a program, in which almost the entire running-time is consumed by the writing part. Observe, however, the following particularity of our method. When we search on up to O(B) threefolds, diﬀering only by the values of a and b, simultaneously, then the running-time is still O(B3).

5.4. Remark (Running-times). We worked with B = 5 000 for the cubics and B = 100 000 for the quartics. In either case, we dealt with all threefolds arising for a, b =1,...,100, simultaneously. The by-far largest portion of the running-time was spent on point search on the quartics. All in all, this took around 155 days of CPU time. This is approximately only three times longer than searching on a single threefold had lasted. Searching for conics was done within 32 days. In comparison with this, the corresponding computations for the cubics could be done in a negligible amount of time. The reason for this is simply that for the cubics the search bound was by far lower. A program with integrated line detection took us approximately ten days. To compute Peyre’s constants, the precomputation was actually the main part. Point counting using FFT took 100 hours for the cubics and 100 hours for the quartics. For the ﬁnal computation, the running-time was a quarter of an hour for either exponent.

ii. The results for the cubics.

5.5. We counted all É-rational points of height less than 5 000 on the three- 3 ≤ 3 ∼ 3 folds Xa,b where a, b =1,...,100 and b a. Note that Xa,b = Xb,a. Points lying on one of the elliptic cones or on a sporadic É-rational line in Xa,b were excluded from the count. The smallest number of points found is 3 930 278 for (a, b)=(98, 95). The largest numbers of points are 332 137 752 for (a, b)=(7, 1) and 355 689 300 in the case that a =1and b =1.

Sec. 5] results 201

3 ≤ On the other hand, for each threefold Xa,b whereas a, b =1,...,100 and b +3 a, we calculated the expected number of points and the quotients

# { points of height

Let us visualize the quotients by the two histograms in Figure 1.

250 250

200 200

150 150

100 100

50 50

0 0.88 0.9 0.92 0.94 0.96 0.98 1 1.02 0 0.88 0.9 0.92 0.94 0.96 0.98 1 1.02

Figure 1. Distribution of the quotients for B = 1 000 and B = 5 000.

Table 2. Parameters of the distribution in the cubic case.

B = 1 000 B = 2 000 B = 5 000 mean value 0.981 79 0.988 54 0.993 83 standard deviation 0.012 74 0.008 23 0.004 55

The statistical parameters are listed in Table 2.

iii. The results for the quartics.

5.6. We counted all É-rational points of height less than 100 000 on the 4

threefolds Xa,b,wherea, b =1,...,100. It turns out that on 5 015 of these varieties, É there are no É-rational points occurring at all as the equation is unsolvable in p for some small p. In this situation, Manin’s conjecture is true, trivially.

202 points of bounded height on threefolds [Chap. VI

For the remaining varieties, the points lying on a known É-rational conic in Xa,b were excluded from the count. Table 3 shows the quartics sorted by the numbers of points remaining. The statistical parameters are listed in Table 4.

Table 3. Numbers of points of height < 100 000 on the quartics.

a b # points # not on conic # expected (by Manin and Peyre) 29 29 2 2 135 58 87 288 288 272 58 58 290 290 388 87 87 386 386 357 ...... 34 1 9 938 976 5 691 456 5 673 000 17 64 5 708 664 5 708 664 5 643 000 1 14 7 205 502 6 361 638 6 483 000 3 1 12 657 056 7 439 616 7 526 000

Table 4. Parameters of the distribution in the quartic case.

B = 1 000 B = 10 000 B = 100 000 mean value 0.9853 0.9957 0.9982 standard deviation 0.3159 0.1130 0.0372

We see that the variation of the quotients is higher than in the cubic case. This is also noticeable from the histograms shown in Figures 2 and 3. iv. Interpretation of the result.

5.7. The results suggest that Manin’s conjecture should be true for the two families of threefolds considered. In the cubic case, the standard deviation is far smaller than in the case of the quartics. This, however, is not very surprising as on a cubic there tend to be many more rational points than on a quartic. This makes the sample more reliable.

5.8. Remark. The data we collected might be used to test the sharpening of the asymptotic formula (∗) suggested by Sir Peter Swinnerton-Dyer [SD05].

5.9. Question. Our calculations seem to indicate that the number of rational points often approaches its expected value from below. Is that more than an accidental eﬀect?

Sec. 5] results 203

0 0.7 0.8 0.9 1 1.1 1.2 1.3 Figure 2. Distribution of the quotients for B = 100 000.

7 7

6 6

5 5

4 4

3 3

2 2

1 1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Figure 3. Distribution of the quotients for B = 1 000 and B = 10 000.

CHAPTER VII

On the smallest point on a diagonal cubic surface∗ All life is an experiment. The more experiments you make the better. Ralph Waldo Emerson (1842)

1. Introduction

É ∅ 1.1. Let X ⊆ P be a Fano variety deﬁned over É.IfX( ) = for every

É ν

∈ É ∅ ν Val( É), then it is natural to ask whether X( ) = . This is just the classical Hasse principle. Adopting a more quantitative point of view, it would be desirable

to have an a priori upper bound for the height of the smallest É-rational point ∅ on X. This would, in fact, allow us to effectively decide whether X( É) = or not. When X is a conic, Legendre’s theorem on zeroes of ternary quadratic forms proves the Hasse principle and, moreover, it yields an effective bound for the smallest point. For quadrics of arbitrary dimension, the same is true by an observation due to J. W. S. Cassels [Cas55]. Further, there is a theorem of C. L. Siegel [Si69, Satz 1], which provides a generalization to hypersurfaces defined by norm equations. For more general Fano varieties, however, there is no theoretical upper bound

known for the height of the smallest É-rational point. Some of these varieties fail the Hasse principle.

1.2. The conjecture of Manin states that the number of É-rational points of anticanonical height

might lead to the expectation that m(X), the smallest height of a É-rational point C on X,isalwayslessthan τ(X) for a certain absolute constant C.

1.3. To test this expectation, we computed the Tamagawa number and

ascertained the smallest É-rational point for each of the cubic surfaces given by

ax3 + by3 +2z3 + w3 =0 for a =1, ... ,3000 and b =1, ... ,300.

∗This chapter is a revised and extended version of the article, On the smallest point on a diagonal cubic surface, Experimental Mathematics 19 (2010), 181–193, with A.-S. Elsenhans and the author.

205

206 on the smallest point on a diagonal cubic surface [Chap. VII

Thereby, we restricted our considerations to the cases in which i) a and b are odd, ii) there exists an odd prime p dividing a but not b such that 3 νp(a),or iii) there exists an odd prime p dividing b but not a such that 3 νp(b). This guarantees that we are in the “first case”, according to the classification of J.-L. Colliot-Thélène, D. Kanevsky, and J.-J. Sansuc [CT/K/S], cf. Section IV.6. In this case, the effect of the Brauer–Manin obstruction is clear. Precisely two thirds of the adelic points are excluded. In addition, we assume that a>b+3. The inequality a ≥ b is necessary in order to avoid duplications. Further, surfaces such that |a − b|≤3 trivially have rational points of uncharacteristically small height.

The results are summarized in Figure 1 below.

Figure 1. Height of smallest point versus Tamagawa number.

Actually, the sample described consists of 849 781 surfaces. Among them, 802 891 turn out to have rational points. Each such surface is marked at its proper place (τ(S), m(S)) in the diagram. It is apparent from the diagram that the experiment agrees with the expectation above. The slope of a line tangent to the top right of the scatter plot is indeed near (−1). However, we will show in Section 10 that, as in the threefold case, the inequality C m(X) < τ(X) does not hold in general. The following remains a logical possibility.

1.4. Question. For every ε>0, does there exist a constant C(ε) such that, for each cubic surface, C(ε) m(X) < ? τ(X)1+ε

Sec. 1] introduction 207

1.5. Remark. In Chapter VI, we presented some theoretical and experi-

mental results concerning the height of the smallest É-rational point on diagonal

quartic threefolds in P4 . É It will turn out that the analogies to the case treated before are enormous. How- ever, there are a few points indicating that the case of a diagonal cubic surface is technically more complicated. An obvious diﬀerence is that the Picard rank of a quartic threefold is always

1 É

equal to one. By consequence, H Gal( É/ ), Pic(X ) is automatically the triv- É ial group. Both these observations are wrong, in general, for diagonal cubic surfaces. Hence, the factors α and β in the definition of Peyre’s constant (Definition II.7.1) are not always the same and need to be considered. Further, one has to expect that a Brauer–Manin obstruction is present. Finally, the L-series at 1 is only conditionally convergent. Thus, it cannot be computed naively, just by taking its first few summands.

1.6. Plan of this chapter. Sections 2 through 8 will be devoted to the computations, which led to the diagram shown in Figure 1. Recall that Peyre’s constant τ(S) was discussed in a very general setting in Sec- tions II.5 through II.7. Nevertheless, it seems that the concrete situation considered in this chapter is, in many aspects, a lot simpler than the general one. Thus, we think it is wise to recap the deﬁnition of τ(S) in a compressed form, which is suitable for the needs for the present problem. This will be done in Section 2 in the hope that it makes this chapter accessible even for those readers who mainly ignored about Chapter II. The next ﬁve sections will discuss the factors this constant is composed of. First we will show that α and β are bounded. Then, in Section 4, we will use the Lefschetz trace formula in order to estimate the product over all local factors τp(S) at the good primes p, uniformly over all cubic surfaces. Sections 5 and 6 will be concerned with the L-factors. In Section 5, for diagonal cubic surfaces, we will give an explicit decomposition of the Galois representation

⊗ 

Pic(S) into irreducible components. As an application of this, we eﬃciently computed the values of the corresponding Artin L-functions at 1. Our method will be presented in detail in Section 6. In Section 7, we will explain our approach to dealing with the factor τ∞(S). This requires numerical integration. Further, we will describe our computations of the Euler products over all non-Archimedean primes. Finally, in Section 8, we will describe our method to ﬁnd the smallest point on every surface in the sample. Sections 9 and 10 will be more theoretical in nature. We will construct a se- { (q)} (q) (q) quence S q∈Æ of diagonal cubic surfaces such that m(S )τ(S ) is unbounded, 1 and we will prove an estimate from below for τ(X) .

\{ } 4 1.7. Notation. Let a =(a0, ... ,a3) ∈ ( 0 ) be a vector. Then, we denote by Sa the cubic surface in P3 given by a x3 + ... + a x3 =0. É 0 0 3 3

208 on the smallest point on a diagonal cubic surface [Chap. VII

2. Peyre’s constant

2.1. Recall from Deﬁnition II.7.1 that E. Peyre’s Tamagawa-type number was deﬁned as

· · t Br − · τ(X):=α(X) β(X) lim (s 1) L(s, χPic(X )) τH X( É ) s→1 É for t = rk Pic(X). The factor β(X) is simply

1 É

β(X):=#H Gal( É/ ), Pic(X ) .

É ⊂ ⊗ Ê

Further, α(X) is given as follows. Let Λeﬀ (X) Pic(X) be the cone generated

⊗ t Ê Ê by the eﬀective divisors.∼ Identify Pic(X) with via a mapping induced by

= t ∨ t ∨

−→ ⊂ Ê an isomorphism Pic(X) . Consider the dual cone Λeﬀ (X) ( ) .Then · { ∈ ∨ | − ≤ } α(X):=t vol x Λeﬀ x, K 1 .

· É

L( ,χPic(X )) denotes the Artin L-function of the Gal( É/ )-representation

É ⊗ 

Pic(X ) , which contains the trivial representation t times as a direct sum- É t

mand. Therefore, L(s, χPic(X ))=ζ(s) · L(s, χP ) and É

t lim (s − 1) L(s, χPic(X ))=L(1,χP ), s→1 É where ζ denotes the Riemann zeta function and P is a representation not containing trivial components. [Mu-y, Corollary 11.5 and Corollary 11.4] show that L(s, χP )

has neither a pole nor a zero at s =1.

Finally, τH is the Tamagawa measure on the set X( É ) of adelic points on X,and

⊆ É X( É ) X( ) denotes the part that is not aﬀected by the Brauer–Manin obstruction.

2.2. As X is projective, we have

X( É )= X( ν ). ∈ É ν Val() τH is deﬁned to be a product measure τH := τν . ν∈Val( É)

For a prime number p, the local measure τp on X( Ép) is given as follows.

Let X ⊆ P3 be the model of X given by the deﬁning cubic equation. For

∈ k a X( /p ), put

(k) k

{ ∈ É | ≡ } U a := x X( p) x a (mod p ) .

Then

(k) −1 Ip τp(U ):=det(1− p Frobp | Pic(X ) ) a É

m k

| ≡ } #{ y ∈ X( /p ) y a (mod p ) · lim . m→∞ pm dim X

Here, Pic(X )Ip denotes the ﬁxed module under the inertia group. É

Sec. 3] the factors α and β 209

τ∞ is described in Deﬁnition II.6.11. In the case of a cubic surface deﬁned by the equation f =0, this yields 1 τ∞(U)= ω 2 Leray CU

|x0|,... ,|x3|≤1 Ê for U ⊂ X( Ê). Here, ωLeray is the Leray measure on the cone CX( ) associated to the equation f =0. The Leray measure is related to the usual hypersurface measure by the formula 1 ωLeray = grad f ωhyp.

2.3. The results. In the case of diagonal cubic surfaces, there is an estimate 1 for m(X) in terms of τ(X).Namely,τ(X) admits a fundamental ﬁniteness property. More precisely, in Section 9, we will show the following theorem.

\{ } 4 a Theorem. Let a =(a0, ... ,a3) ∈ ( 0 ) be a vector. Denote by X the cubic surface in P3 given by a x3 + ... + a x3 =0. Then, for each ε>0,thereexists É 0 0 3 3 a constant C(ε) > 0 such that

1 1 −ε ≥ C(ε) · H 1 : ... : 1 3 . τ(Xa) a0 a3

Corollary (Fundamental ﬁniteness). For each T>0, there are only ﬁnitely many diagonal cubic surfaces Xa : a x3 + ... + a x3 =0in P3 such that τ(V a) >T. 0 0 3 3 É

Corollary (An ineﬃcient search bound). There exists a monotonically decreasing function F :(0, ∞) → [0, ∞),thesearch bound, satisfying the following condition.

a 3 3 Let X be the cubic surface given by the equation a0x0 + ... + a3x3 =0.As-

a a a

 ∅ É ≤ sume X ( É) = .ThenX admits a -rational point of height F (τ(X )).

Proof. One may simply put F (t):= max min H(P ). τ(Xa)≥t a P ∈X ( É) a ∅ X (É)=

a a a ∅ In other words, we have m(X ) ≤ F (τ(X )) as soon as X ( É) = .

3. The factors α and β

3.1. Recall that on a smooth cubic surface X over an algebraically closed 7

ﬁeld there are exactly 27 lines. For the Picard group, which is isomorphic to , the classes of these lines form a system of generators.

210 on the smallest point on a diagonal cubic surface [Chap. VII

3.2. Notation. i) The set L of the 27 lines is equipped with the intersection product , : L×L →{−1, 0, 1}. The pair (L , , ) is the same for all smooth cubic surfaces. It is well known from Chapter III.8.21 that the group of permutations of L respecting , is isomorphic to W (E6). We ﬁx such an isomorphism.

Denote by F ⊂ Div(X ) the group generated by the 27 lines and by F0 ⊂ F the subgroup of principal divisors. Then F is equipped with an operation of W (E6) ∼

such that F0 is a W (E6)-submodule. We have Pic(X ) = F/F0.

É É ii) If X is a smooth cubic surface over É,thenGal( / ) ∼operates canonically on =

the set LX of the 27 lines on X . Fix a bijection iX : LX −→ L respecting the in-

É → tersection pairing. This induces a group homomorphism ιX : Gal( É/ ) W (E6). We denote its image by G ⊆ W (E6).

3.3. Lemma. There is a constant C such that, for all smooth cubic sur-

faces X over É, 1 ≤ β(X) ≤ C.

1 É Proof. By deﬁnition, β(X)=#H Gal( É/ ), Pic(X ) . Using the notation just É

1 1 É

introduced, we may write H Gal( É/ ), Pic(X ) = H (G, F/F0). É Note that this cohomology group is always ﬁnite. Indeed, since G is a ﬁnite group

and F/F0 is a finite [G]-module, the description via the standard complex shows it is finitely generated. Further, it is annihilated by #G. 1 H (G, F/F0) depends only on the subgroup G ⊂ W (E6) occurring. For that, there are finitely many possibilities. This implies the claim.

3.4. Remark. A more precise consideration (cf. Proposition III.8.18) yields a canonical isomorphism

1 ∼ ∩ É É

H Gal( É/ ), Pic(X ) = Hom (NF F0)/NF0, / . É

Here, N is the norm map under the operation of G. As an application of this, one may inspect the 350 conjugacy classes of subgroups of W (E6) using GAP. See Chapter III.8.23 or the complete list of the values of

1 É

H Gal( É/ ), Pic(X ) given in the Appendix. The calculations show that the É lemma is actually true for C =9.

3.5. Lemma. There are positive constants C1 and C2 such that, for all

É ∅

smooth cubic surfaces X over satisfying X( É ) = ,

C1 ≤ α(X) ≤ C2 .

Proof. Again, we claim that α(X) is completely determined by the group G ⊆ W (E6). Thus, suppose that we do not have the full information available about what surface X is but are given the group G only.

Sec. 4] a technical lemma 211

∼ G

 ∅

The assumption X( É ) = makes sure that Pic(X) = Pic(X ) [K/T,Re- É ∼ G mark 3.2.ii)]. We may therefore write Pic(X) = (F/F0) . The eﬀective cone

∼ G

⊂ ⊗ ⊗ Λeff (X) Pic(X) = (F/F0) is generated by the symmetrizations of the classes 1, ... ,27 of the 27 lines in F . In particular, it is determined by G, com- − 1 pletely. Further, we have K = 9 (1 + ... + 27). These data are sufficient to compute α(X) according to its very definition.

3.6. Remark. Recent computations [D/E/J] show that one may actually 1 choose the values C1 := 120 and C2 := 2.

3.7. Remark. In the experiment, we work entirely with cubic surfaces of

Picard rank one over É. This may easily be seem from Theorem 5.6 below. There- fore, we always have α(X)=1. Further, Theorem IV.6.4.a) implies that β(X)=3.

4. A technical lemma

4.1. Sublemma. a) (Good reduction)

(a0,... ,a3) n 2n If p 3a0 · ... · a3, then the sequence #X ( /p )/p is constant. n∈Æ b) (Bad reduction)

(a0,... ,a3) n 2n i) If p divides a0 · ...·a3 but not 3, then the sequence #X ( /p )/p n∈Æ n becomes stationary as soon as p does not divide any of the coeﬃcients a0,...,a3.

(a0,... ,a3) n 2n ii) If p =3, then the sequence #X ( /p )/p becomes stationary

n∈Æ n as soon as 3 does not divide any of the numbers 3a0,...,3a3.

4.2. Lemma. There are two positive constants C1 and C2 such that for \{ } all a0,...,a3 ∈ 0 (a0, ... ,a3) C1 < τp X ( Ép)

(a0,... ,a3) − #X ( p) (a0,... ,a3) − 1 | · τp X ( Ép) =det 1 p Frobp Pic(X ) . É p2 Further, for the number of points on a non-singular cubic surface over a ﬁ-

nite ﬁeld, the Lefschetz trace formula can be made completely explicit [Man,

Ðº ÌÓÖÑ º½ IV, ]. It shows (a0,... ,a3) 2 · |

#X ( p)=p + p tr Frobp Pic(X ) +1. É

212 on the smallest point on a diagonal cubic surface [Chap. VII

Denoting the eigenvalues of the Frobenius on Pic(X ) by λ1, ... ,λ7, we ﬁnd É (a0,... ,a3) τp X ( Ép)

−1 −1 −1 =(1− λ1p )(1 − λ2p ) · ... · (1 − λ7p ) −1 −2 · [1+(λ1 + ··· + λ7)p + p ]

−1 −2 −7 −1 −2 =(1− σ1p + σ2p ∓ ... − σ7p )(1 + σ1p + p )

− 2 −2 − − −3 =1+(1 σ1 + σ2)p (σ1 σ1σ2 + σ3)p −7 −8 −9 ± ... − (σ5 − σ1σ6 + σ7)p +(σ6 − σ1σ7)p − σ7p , where σi denote the elementary symmetric functions in λ1,...,λ7. | | | |≤ 7 ≤ j We know λi =1for all i. Estimating very roughly, we have σj ( j ) 7 and see − − − − 2 3 7 9 (a0, ... ,a3) 1 99p − 7·99p − ... − 7 ·99p ≤ τp X ( Ép) ≤ 1+99p−2 +7·99p−3 + ... +77 · 99p−9 . − − − 2 1 (a0,... ,a3) 2 1 I.e., 1 99p 1−7/p <τp X ( Ép) < 1+99p 1−7/p . The inﬁnite product − −2 1 −2 1 over all 1 99p 1−7/p (respectively 1+99p 1−7/p )isconvergent. The left-hand side is positive for p>13. For the small primes remain-

ing, we need a better lower bound. For this, note that a cubic surface over a ﬁnite ﬁeld p always has at least one p-rational point. This yields (a0, ... ,a3) ≥ − 7 2 τp X ( Ép) (1 1/p) /p > 0.

4.3. Remark. The correctional factors −1 Ip

det 1 − p Frobp | Pic(X ) É are all positive. Indeed, for a pair of complex conjugate eigenvalues, we have (1 − λp−1)(1 − λp−1)=|1 − λp−1|2 > 0, and an eigenvalue of 1 or (−1) contributes a factor 1 ± p−1 > 0. Consequently, we always have 1 7 1 7 −1 Ip 1 − < det 1 − p Frobp | Pic(X ) < 1+ . p É p

5. Splitting the Picard group

5.1. In the case of the diagonal cubic surface X(a0,... ,a3) ⊂ P3 given É

3 3 (a0,... ,a3)

∈ \{ } by a0x0 + ... + a3x3 =0for a0, ... ,a3 0 ,the27 lines on X

2 may easily be written down explicitly. Indeed, for each pair (i, j) ∈ ( /3 ) , the system 3 a x + ζi 3 a x =0, 0 0 3 1 1 3 j 3 a2 x2 + ζ3 a3 x3 =0

Sec. 5] splitting the picard group 213 of equations defines a line on X(a0,... ,a3). Decomposing the index set {0,...,3} differently into two subsets of two elements each yields all the lines. In particular, 3 3 3 we see that the 27 lines may be defined over K = É ζ3, a1/a0, a2/a0, a3/a0 .

5.2. Fact. Let p be a prime number, and let a0, ... ,a3 be integers not divisible by p.Then

(a0,... ,a3) #X ( p) ⎧ 2 2 2 2 2 ⎪p + 1+χ3(a0a1a2a3)+χ3(a0a1a2a3) ⎨ 2 2 2 2 + χ3(a0a1a2a3)+χ3(a0a1a2a3) = ⎪ + χ (a a2a2a )+χ (a2a a a2) p +1 if p ≡ 1(mod 3) , ⎩⎪ 3 0 1 2 3 3 0 1 2 3 p2 + p +1 if p ≡ 2(mod 3) .

≡ ∗ → Here, in the case p 1(mod 3), χ3 : p denotes a cubic residue character. Proof. If p ≡ 2(mod 3), then every residue class modulo p has a unique cubic root.

(a0,... ,a3) 2

→ → Therefore, the map X ( p) P ( p) given by (x : y : z : w) (x : y : z) (a0,... ,a3) 2 is bijective. This shows #X ( p)=p + p +1. Turn to the case p ≡ 1(mod 3). It is classically known that, on a degree m ≡ diagonal variety, the number of p-rational points for p 1(mod m) may be determined using Jacobi sums. The formula given follows immediately from [I/R, 2 Chapter 10, Theorem 2] together with the well-known relation g(χ3)g(χ3)=p for cubic Gauß sums.

\{ } 5.3. Lemma. Let a0, ... ,a3 ∈ 0 . Then, for each prime p such that p 3a0 · ... · a3, (a ,...,a ) (a ,...,a ) 0 3

0 3 | ⊗ 

χPic(X )(Frobp)= tr Frobp Pic(X ) É É ⎧ ⎪ 2 2 2 2 ⎪ χ3(a0a1a2a3)+χ3(a0a1a2a3) ⎨ 2 2 2 2 + χ3(a0a1a2a3)+χ3(a0a1a2a3) = 2 2 2 2 ≡ ⎪+ χ3(a0a1a2a3)+χ3(a0a1a2a3)+1 if p 1(mod 3) , ⎩⎪ 1ifp ≡ 2(mod 3) .

(a0,...,a3) ⊗ 

Proof. As we have good reduction, the trace of Frobp on Pic(X ) is

(a0,...,a3) É ⊗ 

thesameasthatofFrobonPic(X ) . Further, for the number of points

on a non-singular cubic surface over a ﬁnite ﬁeld, the Lefschetz trace formula can

Ðº ÌÓÖÑ º½ be made completely explicit [Man, IV, ]. It shows

(a0,... ,a3) 2 (a0,... ,a3)

 · ⊗ 

#X ( p)=p + p tr Frob Pic(X ) +1. p The explicit formulas for the numbers of points given in Fact 5.2 therefore yield the assertion. √ 3

5.4. Notation. For A an integer, denote the ﬁeld É(ζ3, A) by K.

∗

É → ⊂ Further, let G := Gal(K/É), H := Gal(K/ (ζ3)),andletχ: H ζ3 K G be a primitive character. Then we write ν := indH (χ) for the induced character and VK for the corresponding G-representation.

214 on the smallest point on a diagonal cubic surface [Chap. VII

If K is of degree three over É(ζ3),thenV is an irreducible rank two representation

∼ K ∼

 ⊕ of G = S3.Otherwise,K = É(ζ3).ThenV = M splits into the direct sum

∼ of a trivial and a non-trivial one-dimensional representation of H = /2 .

K É We will freely consider V as a Gal( É/ )-representation.

5.5. Lemma. Let A be any integer. Then, for a prime p not dividing A, we have √ 3 χ (A)+χ (A)ifp ≡ 1(mod 3) , ν É(ζ3, A)(Frob )= 3 3 p 0ifp ≡ 2(mod 3) . Proof. The primitive character is unique up to conjugation by an element of G. Therefore, the induced character λ is well deﬁned. The Kummer pairing allows us to make a deﬁnite √ choice√ for χ as follows. Fix an

3 3 → embedding σ : É(ζ3) . Then put χ(g):=σ g( A)/ A .

If p ≡ 2(mod 3),thenp remains prime in É(ζ3). This means, Frobp acts non- ∈ \ trivially on É(ζ3), i.e., Frobp G H.SinceH is a normal subgroup in G,the induced character vanishes on such an element. ≡ For p 1(mod 3),wehavethat(p) splits in É(ζ3).Letuswrite(p)=pp.

∗ → The choice of p is equivalent to the choice of a homomorphism ι: ζ3 p . The Frobenius Frobp is determined only up to conjugation, we may choose Frobp √= Frobp ∈ H. Then, directly by the deﬁnition of an induced character, 3

É(ζ3, A) ν (Frobp)=χ(Frobp)+χ(Frobp). We need to show that χ(Frobp)=χ3(A) or χ(Frobp)=χ3(A). For this, by the choice made above, we have √ √ 3 3 χ(Frob p):=σ Frobp( A)/ A .

After reduction modulo p,wemaywrite

√ √ √ √ − 3 3 3 p 3 p 1 Frob( A)/ A =( A) / A = A 3 .

√ √ − 3 3 −1 p 1 Therefore, Frobp( A)/ A = ι (A 3 ),whichshows

− −1 p 1 χ(Frobp)=σ ι (A 3 ) .

That ﬁnal formula is a deﬁnition for a cubic residue character at A.

\{ } É É 5.6. Theorem. Let a0,...,a3 ∈ 0 . Then the Gal( / )-representa-

(a0,...,a3) ⊗ 

tion Pic(X ) splits into the direct sum É

(a0,...,a3) ⊗ ∼ K1 K2 K3 ⊕ ⊕ ⊕

Pic X = V V V , É

3 2 2 3 2 2 É where K1,K2,andK3 denote the ﬁelds É(ζ3, a0a1a2a3), (ζ3, a0a1a2a3), 3 2 2 and É(ζ3, a0a1a2a3), respectively. Proof. We will show that the representations on both sides have the same character. For that, by virtue of the Chebotarev density theorem, it suﬃces to consider the values at the Frobenii Frobp for p 3a0 · ... · a3.

Sec. 5] splitting the picard group 215

For the representation on the left-hand side, χPic(X(a0,...,a3))(Frobp) has been computed in Lemma 5.3. For the representation on theÉ right-hand side, Lemma 5.5 shows that exactly the same formula is true.

\{ } 5.7. Corollary. Let a0,...,a3 ∈ 0 be integers,

(a0,... ,a3) ⊗ 

P := Pic X ,

É É considered as a Gal( É/ )-representation, and denote by χP be the associ-

3 2 2 3 2 2 É ated character. Put K1 := É(ζ3, a0a1a2a3), K2 := (ζ3, a0a1a2a3),and 3 2 2 K3 := É(ζ3, a0a1a2a3). Then, for the Artin conductor NχP of χP , we have

2 D D D − NχP = (K1) (K2) (K3)/( 27) ,

where É Disc(K/É)if[K : (ζ3)] = 3 , D(K):= − 27 if K = É(ζ3) .

2 − É Proof. We have to show NνK = D(K)/( 3). Assume ﬁrst that [K : (ζ3)] = 3. Then the conductor-discriminant formula [Ne, Chapter VII, Section (11.9)] shows

É − É É Disc(K/ )=N NM NνK and 3=Disc( (ζ3)/ )=N NM ,whichto- K ⊕ gether yield the assertion. In the opposite case, we have V = M and

K − Nν = N NM = 3.

5.8. Lemma. Let a and b be integers diﬀerent from zero. Then √

3 2 ≤ 9 4 4 É Disc É(ζ3, ab )/ 3 a b .

Proof. We have, at ﬁrst, √ √

3 2 ≤ 3 3 2 2 É É É · É É Disc É(ζ3, ab )/ Disc (ζ3)/ Disc ( ab )/ √

3 2 2 É =27· Disc É( ab )/ .

Further, by [Mc, Chapter 2, Exercise 41], we know √

3 2 ≤ 3 2 2 É Disc É( ab )/ 3 a b .

This shows √

3 2 ≤ 9 4 4 É Disc É(ζ3, ab )/ 3 a b .

\{ } 5.9. Corollary. Let a0,...,a3 ∈ 0 be integers,

(a0,... ,a3) ⊗ 

P := Pic X ,

É É considered as a Gal( É/ )-representation, and let χP be the associated character.

Then, for the Artin conductor NχP of χP , we have the estimate

| |≤ 12 · · 6 NχP 3 (a0 ... a3) .

216 on the smallest point on a diagonal cubic surface [Chap. VII

9 4 Proof. Lemma 5.8 shows | D(Ki)|≤3 (a0 · ...· a3) for i =1, 2,and3. The assertion follows immediately from this.

6. The computation of the L-function at 1

We now return to the particular diagonal cubic surfaces treated in the numerical experiment. Cf. Section 1.3 for a description of our sample.

\{ } 3

6.1. Lemma. For a, b ∈ 0 , consider in P the diagonal cubic surface É S = S(a,b,2,1). Assume that S fulﬁlls conditions 1.3.i), ii),oriii). i) Then, rk Pic(S)=1. ii) Furthermore, there is the relation

K1 K2 K3 lim (s − 1)L(s, χPic(S ))=L(1,ν )L(1,ν )L(1,ν ) s→1 É √ √ √

3 3 2 3 2

É É for K1 = É(ζ3, 4ab), K2 = (ζ3, 2ab ),andK3 = (ζ3, 4ab ). Proof. i) The assumptions imply that 4ab, 2ab2,and4ab2 are three non-cubes.

K1 K2 K3 É In particular, the Gal( É/ )-representations V , V ,andV are irreducible of rank two.

Further, a standard application of the Hochschild–Serre spectral sequence en- É

sures that Pic(S) ⊆ Pic(S )Gal( É/ ) is always a subgroup of ﬁnite index. There-

É É

fore, it suﬃces to verify that rk Pic(S )Gal( É/ ) =1. For this, we note that

É É

Gal( É/ ) ⊗ 

by Theorem 5.6 Pic(S ) splits into a trivial and three irreducible

É É Gal( É/ )-representations.

K1 K2 K3

ii) Note again that χPic(S ) =1+ν + ν + ν . The assertion follows directly É from [Ne, Chapter VII, Theorem (10.4).ii)].

6.2. Observations. i) The character νKi is induced by a non-trivial char-

acter of the group Gal(Ki/É(ζ3)) of order three. Therefore, by [Ne, Chap- ter VII, Theorem (10.4).iv)], we may understand L(s, νKi ) as the Artin L-function

over É(ζ3) associated to that character.

ii) Further, Ki/É(ζ3) is an abelian extension. Then, [Ne, Chapter VII, Theo- rem (10.6)] shows that L(s, νKi ) coincides with the Hecke L-function given by the 2 2 generalized Dirichlet character of order three modulo 4ab, 2ab ,or4ab over É(ζ3). An elementary proof of this fact requires the cubic reciprocity low [I/R].

6.3. Remarks. i) As L(1,νKi ) is not given by an absolutely convergent series, we cannot evaluate it directly. ii) One could apply the analytic class number formula to compute L(1,νKi ). This approach is, however, not practical for half a million L-functions.

6.4. Notation. From now on, we will denote the generalized Dirichlet char-

acter of order three modulo A by νA and its conductor by m ∈ [ζ3].Further,we → É write N : É(ζ3) for the norm map.

Sec. 6] the computation of the l-function at 1 217

6.5. We complete the L-function by putting 2 Λ(s, ν ):=(−3N(m))s/2 Γ(s)L(s, ν ) . A (2π)s A The completed L-function is connected with a theta function via a Mellin transform. One has ∞ dt Λ(s, ν )= f(t) ts/2 , A t 0 where f is the function deﬁned by √ 1 − 2π N(a) t f(t):= ν (a)e |3m| 6 A

a∈[ζ3]

for t>0. The connection to the Hecke theta function associated to [ζ3] and νA is given by 1 √ f(t):= θ(i t, ν ) . 6 A Inspecting the convergence properties of the series, we see that it converges very rapidly for t 0 while convergence is arbitrarily slow for t close to zero. The functional equation z θ(−1/z, ν )= θ(z,ν ) A i A interchanges the ranges of good and bad convergence. Hence, this equation should be used to compute f(t) for t small. To be more precise, we split the half-line [0, ∞) into two parts and write u ∞ dt dt Λ(s, ν )= f(t) ts/2 + f(t) ts/2 . A t t 0 u

Applying the functional equation of the Hecke theta function to the ﬁrst summand yields

: ∞ 4 5 − 1 |3m| 1 s Λ(s, ν )= 2ν (a) e−xx−sdx A 6 A 2πN(a)

a∈[ζ ] 3 2πN(a) √1 |3m| u 4 5 ∞ ; |3m| s + e−xxs−1dx (∗) 2πN(a) √ 2πN(a) |3m| u for each u>0. This is an absolutely convergent inﬁnite series.

6.6. Remark. TheideatoevaluateanL-function at an arbitrary point ∈ ∗ s using a series analogous to ( ) goes back, at least, to A. F. Lavrik [Lav]. De- scriptions of similar methods may also be found in [St75], [Coh00, Section 10.3], and [Dok].

218 on the smallest point on a diagonal cubic surface [Chap. VII

6.7. Remark. The relation of Λ(s, νA) to a theta function is a particular case of the very general [Ne, Chapter VII, Theorem (8.3)]. In comparison with the

general case, many simpliﬁcations do occur, mainly because É(ζ3) is an imaginary − quadratic number ﬁeld of class number 1. Note that É(ζ3) has discriminant ( 3) and precisely six units.

6.8. Remark. In more generality, the functional equation of a Hecke theta function is of the form τ(ν) z θ(−1/z, ν)= θ(z,ν) . N(m) i Here, τ(ν) is the Gauß sum associated to the character ν [Ne, Chapter VII, Deﬁ- nition (7.4)].

In our case, it is immediate from the deﬁnition that τ(νA) is real. Further, [Ne, Chapter VII, Theorem (7.7)] shows that |τ(νA)| = N(m) such that the coeﬃcient of the functional equation is ±1. Actually, the sign is always positive. Indeed, a direct calculation shows

√ ζ 3 (s)=L(s, νA)ζ(s) . É( A)

Further, in the functional equation of the Dedekind zeta function, the sign is always positive [Ne, Chapter VII, Corollary (5.10)].

6.9. Remarks. i) The convergence of the series (∗)isoptimalwhenu is close to 1. Calculations using diﬀerent values of u may be used for checks [Dok]. ii) The number of summands required for a numerical approximation is about C|m|. The constant C depends on the precision required.

6.10. Remark. There are a number of obvious ideas to optimize the computations. i) The summand for a depends only on the ideal (a). Hence, the summands arise in groups of six. We calculate only once for each group. ii) Both integrals depend only on N(a) and |m|. Thus, we evaluate them only once for each pair (N(a), |m|). iii) The computation of the generalized Dirichlet characters νA is sped up using their ∈ multiplicativity in A. For a concrete value a [ζ3], we ﬁrst use Euler’s criterion to compute νp(a) for all prime numbers p less than 3000. Having tabulated these values, the calculation of all the characters νA at a is done rapidly.

Since we are interested in the evaluation of many L-functions at s =1, some more possibilities for optimization do arise. iv) Actually, the ﬁrst integral is the integral exponential function and the second one is just an exponential function. The numerical evaluation of the integral exponential function could be done by a combination of the power series expansion with a continued fraction expansion [P/F/T/V].

Sec. 7] computing the tamagawa numbers 219

However, there is another method which is better. The arguments of the integral exponential function we meet lie in a rather small range. This range was split up into even smaller intervals. On each interval, we used a polynomial approximation.

6.11. We organized the computations as follows. In a ﬁrst step, we enu- merated all the radicands A for which L(1,νA) had to be computed. We sorted the list and eliminated all repetitions. In addition, for each radicand, we stored its prime decomposition for later use. The resulting list consisted of 557 270 radicands. Only 214 285 diﬀerent conductors occurred.

Then, we evaluated L(1,νA) for all the radicands A occurring. We used formula (∗) for u =1and u =1.2. To evaluate the series numerically, we worked with 64-bit hardware floats and used backward summation. The differences between the two results were always negligible. The whole computation of the values of L took around four days on a 2.2 GHz Opteron processor. In Table 1 below, we present a few of the values computed. The first two lines represent the absolutely largest and the absolutely smallest value of L we found. The three other lines all correspond to conductor 5 380 206, which is the largest conductor appearing in our list. For this maximal conductor, we worked in the ∈ ≤ summation with all a [ζ3] such that N(a) 38 276 797. For smaller conductors, according to Remark 6.9.ii) fewer summands were used.

Table 1. Some values of the L-functions at s =1.

Radicand A L(1,νA) using u =1 L(1,νA) using u =1.2 ...using class number formula 166 249 4.419 173 379 082 995 4.419 173 379 082 997 4.419 173 379 082 996 519 114 130 102 044 100 0.596 117 703 616 924 0.596 117 703 616 918 0.596 117 703 616 923 884 079 232 3 586 804 0.888 154 374 767 605 0.888 154 374 767 607 0.888 154 374 767 604 963 111 775 536 227 198 0.946 251 759 020 570 0.946 251 759 020 576 0.946 251 759 020 569 971 686 643 1 072 454 396 1.437 503 627 427 445 1.437 503 627 427 447 1.437 503 627 427 445 188 453 952

7. Computing the Tamagawa numbers

\{ } 3

7.1. Lemma. For a, b ∈ 0 , consider in P the diagonal cubic surface É S = S(a,b,2,1). Assume that S fulﬁlls conditions 1.3.i), ii),oriii). i) Then, α(S)=1and β(S)=3. ii) Furthermore, one has precisely

Br 1

É τ S( É ) = τ S( ) . H 3 H Proof. i) On a cubic surface, the self-intersection number of the canonical divisor K is equal to 3 which is square-free. Therefore, rk Pic(S)=1immediately implies that Pic(S)=K. This is enough to ensure α(S)=1. β(S) can be computed using the method described in Yu. I. Manin’s book [Man, Proposition 31.3]. Let F ⊂ Div(S) the free abelian group over the 27 lines, let F0 ⊂ F be the subset of principal divisors, and let N : F → F be the norm map

220 on the smallest point on a diagonal cubic surface [Chap. VII under the operation of the Galois group G on F . Then, Yu. I. Manin states that

1 ∼

É É

H Gal( É/ ), Pic(S ) = Hom (NF ∩ F0)/NF0, / . É

We have a group G of order 6, 18,or54.If#G =54,thenG decomposes the 27 lines into three orbits of nine lines each. In this case, an easy calculation shows that

∩ ∼ Hom (NF F0)/NF0, É/ = /3 .

The smaller groups might lead to the decomposition types [3,6, 9, 9] or

∩ ∼ [3, 3, 3, 6, 6, 6]. A calculation in GAP shows Hom (NF F0)/NF0, É/ = /3 in these cases, too. ii) This is known by the work of J.-L. Colliot-Thélène, D. Kanevsky, and J.-J. Sansuc [CT/K/S, Proof of Proposition 2]. Cf. Section IV.6, above.

\{ } 7.2. Corollary. For a, b ∈ 0 , consider the diagonal cubic surface S = S(a,b,2,1). Assume that S fulﬁlls conditions 1.3.i), ii),oriii). Then, for E. Peyre’s Tamagawa-type number, one has

· Ê τ(S) = lim (s − 1)L(s, χPic(S )) · τp S( Ép) τ∞ S( ) . s→1 É p prime 7.3. The factor at the inﬁnite place. Since S is a diagonal cubic sur-

face, the projection from the cone CS( Ê) to the (y, z, w)-space is one-to-one. There- fore, √1 1 τ∞ S( Ê) = dy dz dw . 6 3 a (by3 +2z3 + w3)2/3 (y,z,w)∈[−1,1]3 |x(y,z,w)|≤1 Further, we have < < 3 3 3 3 3 3 3 |by +2z + w | 3 b|y| +2|z| + |w| |x(y, z, w)| = ≤ . a a Since |y|≤1, |z|≤1, |w|≤1,anda>b+3, it turns out that the condition | |≤ x(y, z, w) 1 is actually empty. The integral in the formula for τ∞(S( Ê)) depends only on b. We are left with just 300 diﬀerent integrals. A linear substitution leads to 300 integrals of the same function on an increasing sequence of integration domains. Hence, this sequence can be computed incre- mentally. Doing this, the ﬁrst integrals (for b =1, 2,and3) are critical since the integrand is singular in the domain of integration. Thus, they should not be computed naively. We evaluated them using the approach described in [EJ6].

7.4. Computation of the Euler product. By Lemma 4.2, the Euler product is absolutely convergent and, for the relative error, we have the estimate −2 1 1 99/2 1 1 ± 99p · 1 − − 1 ≤ + O 1 − 7/p p3 N log N 2 p≥N p≥N N log N p≡1(mod 3) p≡2(mod 3) if all bad primes are below N. In particular, the approximation by the ﬁnite product over all primes up to 106 leads to a relative error of less than 4·10−6.

Sec. 8] searching for the smallest solution 221

The computation of the Euler products was done according to their deﬁnition. An optimization which is worth a mention is that we ran the outer loop over the prime numbers and the inner loops over a and b. The whole computation of the Euler products took a quarter of an hour.

8. Searching for the smallest solution

8.1. We will now explain how we generated the data for Figure 1. In addition to computing the Tamagawa type numbers, we had to ﬁnd the points of smallest height. I.e., the smallest solutions of the equations

ax3 + by3 +2z3 + w3 =0, where a =1,...,3000 and b =1,...,300 fulfill the conditions formulated in 1.3. We applied a modification of the strategy due to M. Vallino [CT/K/S, p. 79/80]. The algorithms used are slight modifications of Algorithm VI.5.2. We dealt with the decoupling ax3 +2z3 = −by3 − w3.

8.2. Description of the method. i) In a ﬁrst stage, we worked with a search bound of 100 and ran the algorithm simultaneously on all the 900 000 equations for a =1,...,3000 and b =1,...,300. For exactly 69 074 of these equations, no solution was found. Among them, 67 787 fulﬁlled the congruence conditions formulated in 1.3. In this list, there were only a few duplications. 65 314 of the equations obeyed the limitation a>b+3,too. For these, we ran a test for p-adic solvability. It turned out that only 18 424 of the

remaining 65 314 equations were solvable in Ép for every prime p. ii) We executed the second stage with the corresponding pairs. They were read from a ﬁle. The searching algorithm was run separately for each equation. We worked with search bounds of 200, 400,and800 and stopped when a solution was found. Only 113 equations remained unsolved by that stage. iii) In most of these cases, there was a prime p such that 2 is a cubic non-residue modulo p dividing both a and b. This enforces that both z and w must be divisible by p. We used these strong divisibility conditions when working with search bounds of 4000 and 20 000.

8.3. Remark. Actually, in the last stage, there were only three equations remaining for which no solution had been found with a search bound of B = 4000. They are represented by the pairs (a, b) = (2321, 211), (2331, 222),and(2641, 278). The corresponding smallest solutions are (−125 : −884 : 4220 : −211), (−389:64:4033:1813),and(−1023:−458:11 259:−695), respectively.

8.4. Remark. Altogether, there are exactly 849 781 cubic surfaces fulﬁlling the congruence conditions and limitations given in 1.3. It turned out that 46 890 of them are p-adically unsolvable for some prime p ≡ 1(mod 3).Eachofthe

remaining cubic surfaces admits a É-rational point. Thus, there are no counterexamples to the Hasse principle in our sample. This con- ﬁrms a conjecture of J.-L. Colliot-Thélène, cf. [C/S, Conjecture C].

222 on the smallest point on a diagonal cubic surface [Chap. VII

8.5. Remark. It should be noticed that Algorithm VI.5.2 itself would not work very well on this problem, at least not on the ﬁrst stage. The point is that there are some numbers which appear as values of the expressions ax3 +2z3 and (−by3 − w3), many times. Whether we chose one side or the other, we had a hash function which was quite far from being uniform. Our idea for overcoming this diﬃculty was to replace hashing by sorting. We generate sorted lists of all values taken by the expressions on the two sides. We look for coincidences by a procedure similar to a step of Mergesort.

9. The fundamental ﬁniteness property

9.1. In this section, we return to the case of a general diagonal cubic surface X(a0,... ,a3) ⊂ P3 given by a x3 + ... + a x3 =0. Our goal is to establish the esti- É 0 0 3 3 mate for τ (a0,... ,a3) := τ(X(a0,... ,a3)) formulated as the theorem in Subsection 2.3. For this, in the subsections below, we will give an individual estimate for each of the factors occurring in the deﬁnition of τ(X(a0,... ,a3)). i. An estimate for the L-factor.

9.2. Proposition. For each ε>0, there exist positive constants c1 and c2 such that ·| · · |−ε − t ·| · · |ε c1 a0 ... a3 < lim (s 1) L s, χ (a0, ... ,a3)

s→1 Pic(X ) É \{ } 4 for all (a0,...,a3) ∈ ( 0 ) . Here, t = rk Pic(X).

(a0,... ,a3) ⊗ 

Proof. The Galois representation Pic(X ) contains the trivial repre- É sentation t times as a direct summand. Therefore, t L s, χ (a , ... ,a ) = ζ(s) · L(s, χ ),

Pic(X 0 3 ) P É where ζ denotes the Riemann zeta function and P is a representation not containing trivial components. All we need to show is

−ε ε c1 ·|a0 · ... · a3|

L( · ,χP ) is the product of at most three factors of the form L( · ,λ),whereλ is É the non-trivial Dirichlet character of É(ζ3)/ and at most three factors, which are · K Artin L-functions L( ,ν )for K a purely cubic ﬁeld extension of É(ζ3) as above, 3 2 2 say K = É ζ3, a0a1a2a3 .AsL(1,λ) does not depend on a0, ... ,a3, at all, it will suﬃce to show

−ε K ε c1(ε) ·|a0 · ... · a3| 0.

Sec. 9] the fundamental finiteness property 223

K ∼ V is the only irreducible two dimensional representation of Gal(K/É) = S3. For that reason, by virtue of [Ne, Chapter VII, Corollary (10.5)], we have

· · K 2

ζK (s)=ζ É (s) L(s, λ) L(s, ν ) · K 2 = ζ É(ζ3)(s) L(s, ν ) for a complex variable s. It, therefore, suﬃces in our particular situation to estimate the residue ress=1 ζK (s) of the Dedekind zeta function of K. An estimate from above has been given by C. L. Siegel. In view of the analytic class number formula, his [Si69,Satz1]gives

5 res ζK (s)

ε for a certain constant C. The ﬁnal term is less than c2(ε) ·|a0 · ... · a3| for every ε>0. On the other hand, H. M. Stark [St74, formula (1)] shows

−ε/4 res ζK (s) >C(ε)·Disc(K/É) s=1 −ε for every ε>0, which implies res ζK (s) >c1(ε)·|a0 · ... · a3| . s=1 ii. An estimate for the factors at the ﬁnite places.

9.3. Notation. i) For a prime number p and an integer x =0 , we put (p) νp(x) (p)

x := p . Note x =1/ x p for the normalized p-adic valuation. ii) By putting ν(x):=min ξ∈p ν(ξ), we carry the p-adic valuation from p

r x=(ξ mod pr ) over to /p .

r ν(x) r ∗

· ∈ Note that any 0 = x ∈ /p has the form x = ε p ,whereε ( /p ) is a unit. Clearly, ε is unique only in the case ν(x)=0.

∈ Æ ≤ 9.4. Deﬁnition. For (a0,...,a3) ∈ , r ,andν0,...,ν3 r, put

(r) r 4

{ ∈ | Nν0,... ,ν3;a0,... ,a3 := (x0,...,x3) ( /p )

3 3 r

∈ } ν(x0)=ν0,...,ν(x3)=ν3; a0x0 + ... + a3x3 =0 /p .

For the particular case ν0 = ... = ν3 =0, we will write

Z(r) := N (r) . a0,... ,a3 0,... ,0;a0,... ,a3

I.e.,

(r) r ∗ 4 3 3 r

{ ∈ | ∈ } Za0,... ,a3 = (x0,...,x3) [( /p ) ] a0x0 + ... + a3x3 =0 /p .

(r) (r) We will use the notation za0,... ,a3 := #Za0,... ,a3 .

224 on the smallest point on a diagonal cubic surface [Chap. VII

k 9.5. Sublemma. If p |a0,...,a3 and r>kthen we have

(r) 4k · (r−k) z = p z k k . a0,... ,a3 a0/p ,... ,a3/p

3 3 k k · 3 k · 3 Proof. Since a0x0 + ... + a3x3 = p (a0/p x0 + ... + a3/p x3),thereisa surjection (r) −→ (r−k) ι: Z Z k k , a0,... ,a3 a0/p ,... ,a3/p r−k r−k given by (x0, ... ,x3) → (x0 mod p ), ... ,(x3 mod p ) . The kernel of the

r−k r 4

homomorphism of modules underlying ι is (p /p ) .

k 9.6. Lemma. Assume gcdp(a0,...,a4)=p . Then there is an estimate

(r) ≤ 3r+k za0,... ,a4 3p .

Proof. Suppose ﬁrst that k =0. This means, one of the coeﬃcients is prime to p. Without restriction, assume p a0.

r 3 3

∈ For any (x1,x2,x3) ( /p ) , there appears an equation of the form a0x0 = c.It

r ∗ cannot have more than three solutions in ( /p ) . Indeed, for p odd, this follows

r ∗ directly from the fact that ( /p ) is a cyclic group. On the other hand, in the

r ∗ ∼ r−2

× case p =2,wehave( /2 ) = /2 /2 . Again, there are only up to three solutions possible. The general case may now easily be deduced from Sublemma 9.5. Indeed, if k

On the other hand, if k ≥ r, then the assertion is completely trivial since

(r) (r) 4r ≤ 3r+k 3r+k za0,... ,a3 =#Za0,... ,a3

9.7. Remark. The proof shows that in the case p ≡ 2(mod 3) one could reduce the coeﬃcient to 1. Unfortunately, this observation does not lead to a substantial improvement of our ﬁnal result.

≤ 9.8. Lemma. Let r ∈ Æ and ν0,...,ν3 r.Then

(r) r−ν0 r−ν3 z 3ν 3ν · ϕ(p ) · ... · ϕ(p ) #N (r) = p 0 a0,... ,p 3 a3 . ν0,... ,ν3;a0,... ,a3 ϕ(pr)4

3ν0 3 3ν3 3 ν0 3 ν3 3 Proof. As p a0x0 + ... + p a3x3 = a0(p x0) + ... + a3(p x3) ,wehave a surjection (r) (r) π : Z 3ν 3ν −→ N , p 0 a0,... ,p 3 a3 ν0,... ,ν3;a0,... ,a3

ν0 ν3 given by (x0,...,x3) → (p x0,...,p x3).

Sec. 9] the fundamental finiteness property 225

r r νi

→ → For i =0, ... ,3, consider the mapping ι: /p /p , x p x.Ifνi = r, then ι is the zero map. All ϕ(pr)=(p − 1)pr−1 units are mapped to zero. Oth- νi erwise, observe that ι is p :1onto its image. Further, ν(ι(x)) = νi if and only if x is a unit. By consequence, π is (K(ν0) · ... · K(ν3)):1 when we put K(ν) := pν for ν

∈ \{ } 9.9. Corollary. Let (a0, ... ,a3) ( 0 ) . Then, for the local factor (a0,... ,a3) τp X ( Ép) , one has − (a0,... ,a3) − 1 | Ip

τp X ( Ép) =det 1 p Frobp Pic(X ) É (r) − − r r ν0 r ν3 z 3ν 3ν · ϕ(p ) · ... · ϕ(p ) · lim p 0 a0,... ,p 3 a3 . r→∞ p3r · ϕ(pr)4 ν0,... ,ν3=0

Proof. By Remark II.6.8, we have − (a0,... ,a3) − 1 | Ip

τp X ( Ép) =det 1 p Frobp Pic(X ) É r #N (r) · lim ν0,... ,ν3;a0,... ,a3 . r→∞ p3r ν0,... ,ν3=0 Lemma 9.8 yields the assertion.

\{ } 4 9.10. Proposition. Let (a0, ... ,a3) ∈ ( 0 ) . Then, for each ε such 1 that 0 <ε< 3 , one has 7 3 1−ε 1 1 1 (p) (p) (p) (p) ε (a0,... ,a3) ≤ · · 3 τ X ( É ) 1+ 3 a a a a . p p − 1 − 1 0 1 2 3 p 1 p1−3ε 1 pε Proof. We use the formula from Corollary 9.9. By Remark 4.3, the ﬁrst factor is at most (1+1/p)7. Further, by Lemma 9.6,

(r) 3r ≤ 3ν0 3ν3 z 3ν 3ν /p 3gcdp(p a0,...,p a3) p 0 a0,... ,p 3 a3 3ν0 (p) 3ν3 (p) =3gcdp a0 ,...,p a3 . (p) Writing ki := νp(ai)=νp ai ,wesee

(r) 3r 3ν0+k0 3ν3+k3 z 3ν 3ν /p ≤ 3gcd(p ,...,p ) p 0 a0,... ,p 3 a3 { } =3pmin 3ν0+k0,... ,3ν3+k3 .

1−ε 1−ε We estimate the minimum by a weighted arithmetic mean with weights 3 , 3 , 1−ε 3 ,andε,

min{3ν0 + k0,...,3ν3 + k3} 1 − ε 1 − ε ≤ · (3ν0 + k0)+ · (3ν1 + k1) 3 1 − ε 3 + · (3ν2 + k2)+ε(3ν3 + k3) 3 1 − ε =(1− ε)(ν + ν + ν )+3εν + (k + k + k )+εk . 0 1 2 3 3 0 1 2 3

226 on the smallest point on a diagonal cubic surface [Chap. VII

This shows

(r) − 1−ε 3r (1 ε)(ν0+ν1+ν2)+3εν3+ 3 (k0+k1+k2)+εk3 z 3ν 3ν /p ≤ 3p p 0 a0,... ,p 3 a3 − − (p) (p) (p) 1 ε (p) ε (1 ε)(ν0+ν1+ν2)+3εν3 · 3 =3p a0 a1 a2 a3 .

We may therefore write 7 1−ε 1 (p) (p) (p) (p) ε (a0,... ,a3) ≤ · 3 τ X ( É ) 1+ 3 a a a a p p p 0 1 2 3 r − − − p(1 ε)(ν0+ν1+ν2)+3εν3 · ϕ(pr ν0 ) · ... · ϕ(pr ν3 ) · lim . r→∞ ϕ(pr)4 ν0,... ,ν3=0 Here, the term under the limit is precisely the product of three copies of the ﬁ- nite sum − r p(1−ε)ν · ϕ(pr−ν) r 1 1 p 1 = + ϕ(pr) (pε)ν p − 1 (pε)r ν=0 ν=0 and one copy of the ﬁnite sum − r p3εν · ϕ(pr−ν) r 1 1 p 1 = + . ϕ(pr) (p1−3ε)ν p − 1 (p1−3ε)r ν=0 ν=0 For r →∞, geometric series do appear while the additional summands tend to zero.

9.11. Remark. Unfortunately, the constants 7 3 (ε) 1 · 1 1 Cp := 1+ 3 1 1 p 1 − − 1 − p1 3ε pε (ε) have the property that the product p Cp diverges. On the other hand, we have (ε) (ε) (ε) at least that Cp is bounded for p →∞,sayCp ≤ C .

9.12. Lemma. Let C>1 be any constant. Then, for each ε>0, one has C ≤ c · xε p prime p|x for a suitable constant c (depending on ε).

Proof. This follows directly from [Nat, Theorem 7.2] together with [Nat, Sec- tion 7.1, Exercise 7].

1 9.13. Proposition. For each ε such that 0 <ε< 3 , there exists a constant c such that 1 − ε 1 −ε (a0,... ,a3) ≤ ·| · · | 3 8 · 3 τp X ( Ép) c a0 ... a3 min ai p i=0,... ,3 p prime p prime \{ } 4 for all (a0,...,a3) ∈ ( 0 ) .

Sec. 9] the fundamental finiteness property 227

Proof. The product over all primes of good reduction is bounded by virtue of Lemma 4.2.a). It, therefore, remains to show that 1 − ε 1 −ε (a0,... ,a3) ≤ ·| · · | 3 8 · 3 τp X ( Ép) c a0 ... a3 min ai p . i=0,... ,3 p prime p prime | p 3a0 ... a3 For this, by Proposition 9.10, we have at ﬁrst (p) (p) (p) 1 − ε (p) 3 (a0,... ,a3) ≤ (ε) · 3 4 · 4 ε τp X ( Ép) Cp a0 a1 a2 (a3 ) (p) (p) (p) (p) 1 − ε (p) − 1 (ε) · 3 4 · 3 +ε = Cp a0 a1 a2 a3 (a3 ) .

Here, the indices 0,...,3 are interchangeable. Hence, it is even permissible to write (p) (p) (p) (p) 1 − ε (p) − 1 +ε (a0,... ,a3) ≤ (ε) · 3 4 · 3 τp X ( Ép) Cp a0 a1 a2 a3 max a i i 1 − ε 1 −ε (ε) · (p) (p) (p) (p) 3 4 · 3 = Cp a0 a1 a2 a3 min ai p . i

Now, we multiply over all prime divisors of a0 · ...· a3. Thereby, on the right-hand side, we may twice write the product over all primes since the two rightmost factors are equal to one for p 3a0 · ... · a3,anyway. (a0,... ,a3) τp X ( Ép) p prime | p 3a0 ... a3 1 − ε 1 −ε ≤ (ε) · (p) (p) (p) (p) 3 4 · 3 Cp a0 a1 a2 a3 min ai p i=0,... ,3 p prime p prime p prime p|3a ... a 0 3 1 − ε 1 −ε (ε) ·| · · | 3 4 · 3 = Cp a0 ... a3 min ai p i=0,... ,3 p prime p prime p|3a ... a 0 3 (p) | | (ε) ≤ (ε) when we observe that p a = a .Further,wehaveCp C and, by Lemma 9.12, (ε) ε C ≤ c ·|3a0 · ... · a3| 8 . p prime | p 3a0 ... a3 ε We ﬁnally estimate 3 8 by a constant. The assertion follows. iii. An estimate for the factor at the inﬁnite place.

\{ } 9.14. Corollary. Let a ,...,a ∈ Ê 0 .Then 0 4 3 5 (a0, ... ,a3)

Ê 1 ωCX ( ) = dx ∧ dx ∧ dx . Leray | | 2 1 2 3 3 a0 x0 4

Proof. We apply Lemma II.6.15 to U = Ê and

3 3 f(x0,...,x3):=a0x0 + ... + a3x3.

{ (a0,... ,a3) | } Note that (x0,... ,x3) ∈ CX ( Ê) x0 =0 is a zero set according to the Leray measure as it is for the hypersurface measure.

228 on the smallest point on a diagonal cubic surface [Chap. VII

\{ } 9.15. Lemma. Let a0,...,a3 ∈ Ê 0 .Then (1, ... ,1)

1 CX ( Ê) (a0,... ,a3) τ∞ X ( Ê) = ω . 3 Leray 2 | a0 · ... · a3| (1, ... ,1)

√CX ( Ê) √ 3 3 |x0|≤ |a0|,... ,|x3|≤ |a3| (a0,... ,a3) Proof. According to the deﬁnition of τ∞ X ( Ê) and the corollary above, we need to show 1 1 dx ∧dx ∧ dx | | 2 1 2 3 6 a0 x0 (a0, ... ,a3) CX ( Ê) | |≤ | |≤ x0 1,... , x3 1 1 1 = dX1 ∧ dX2 ∧ dX3 . 3 2 6 | a0 · ... · a3| X0 (1, ... ,1)

√CX ( Ê) √ 3 3 |X0|≤ | a0|,... ,|X3|≤ | a3|

(a0,... ,a3) (1,... ,1) → Ê For that, consider the linear mapping l : CX ( Ê) CX ( ) given 3 3 by (x0,...,x3) → ( a0x0,..., a3x3).Then 1 3 a a a 1 l∗ dX ∧ dX ∧ dX = 1 2 3 dx ∧ dx ∧ dx . X2 1 2 3 2/3 x2 1 2 3 0 a0 0 This immediately yields the assertion when we take into consideration that orien- tations are chosen in such a way that both integrals are positive.

9.16. Proposition. For real numbers 0

Ê 64 1 b CX ( ) ≤ 3 1 ωLeray 64 + log 3 + 3 ω2 b0 +64b0 log , 3 3 b0 CX(1, ... ,1)(Ê) | x0|≤b0,... ,| x3|≤b3 where ω2 is the two dimensional hypersurface measure on the l3-unit sphere

2 3 || |3 | |3 | |3 } S := { (x1,x2,x3) ∈ Ê x1 + x2 + x3 =1 .

Proof. First step. We cover the domain of integration by 25 sets as follows. We put 4 (1,... ,1) ∈ R0 := [−b0,b0] ∩ CX ( Ê). Further, for each σ S4,weset

4 || |≤···≤| | | |≤ ≤| |} Rσ := { (x0,...,x3) ∈ Ê xσ(0) xσ(3) , xi bi,b0 xσ(3) (1,... ,1)

∩ CX ( Ê) .

= (1, ... ,1) = (1, ... ,1) Ê CX ( Ê) CX ( ) Second step. One has ω ≤ ω for every σ ∈ S4. Rσ Leray Rid Leray

4 4

→ Ê → Consider the map iσ : Ê given by (x0, ... ,x3) (xσ(0), ... ,xσ(3)). (1,... ,1) Since CX ( Ê) is deﬁned by a symmetric cubic form, it is invariant under iσ. We claim that iσ(Rσ) ⊆ Rid.

Sec. 9] the fundamental finiteness property 229

Indeed, let (x0, ... ,x3) ∈ Rσ.Theniσ(x0, ... ,x3)=(xσ(0), ... ,xσ(3)) has the properties | xσ(0)|≤ ... ≤|xσ(3)| and b0 ≤|xσ(3)|.Inordertoshow iσ(x0,...,x3) ∈ Rid, all we need to verify is | xσ(i)|≤bi for i =0,...,3.

For this, we use that the bi are sorted. We have | xσ(3)|≤bσ(3) ≤ b3. Further, | xσ(2)|≤bσ(2) and | xσ(2)|≤|xσ(3)|≤bσ(3), one of which is at most equal to b2. Similarly, | xσ(1)|≤bσ(1), | xσ(1)|≤|xσ(2)|≤bσ(2),and | xσ(1)|≤|xσ(3)|≤bσ(3), the smallest of which is not larger than b1.Fi- nally, | xσ(0)|≤bσ(0), | xσ(0)|≤|xσ(1)|≤bσ(1), | xσ(0)|≤|xσ(2)|≤bσ(2),and | xσ(0)|≤|xσ(3)|≤bσ(3),whichshows| xσ(0)|≤b0.

3 3 (1,... ,1) As x0 + ... + x3 is a symmetric form, the Leray measure on CX ( Ê) is

(1,... ,1) 4 ⊂ Ê invariant under the canonical operation of S4 on CX ( Ê) . This means,

(1, ... ,1) (1, ... ,1) Ê CX ( Ê) CX ( ) ∈ we have (iσ)∗ωLeray = ωLeray for each σ S4. Altogether,

(1, ... ,1) (1, ... ,1) (1, ... ,1) (1, ... ,1)

Ê Ê Ê CX ( Ê) ≤ CX ( ) CX ( ) CX ( ) ωLeray ωLeray = (iσ)∗ ωLeray = ωLeray .

−1 Rσ iσ (Rid) Rid Rid

= (1, ... ,1) √ CX ( Ê) 1 3 Third step. We have ω ≤ 3 ω2b0. R0 Leray 3

By virtue of Corollary 9.14, we have (1, ... ,1) CX ( Ê) 1 1 ∧ ∧ ωLeray = 2 dx0 dx1 dx2 3 x3 R0 R0 1 1 = dx dx dx , 3 3 3 2/3 0 1 2 3 (x0 + x1 + x2) π(R0)

(1,... ,1) 3

→ Ê → where π : CX ( Ê) , (x0,x1,x2,x3) (x0,x1,x2), denotes the projection to the ﬁrst three coordinates.

We enlarge the domain of integration to

3 3 3 3 3

{ ∈ Ê || | | | | | ≤ } R := (x1,x2,x3) x0 + x1 + x2 3b0 .

Then, by homogeneity, we see √ 3 3b0 1 1 √ dx dx dx = ω · · r2 dr = ω · 3 3b . 3 3 3 2/3 0 1 2 2 2 2 0 (x0 + x1 + x2) r R 0

= (1, ... ,1) CX ( Ê) 8 8 8 b1 Fourth step. We have ω ≤ ( + log 3)b0 + b0 log . Leray 3 9 3 b0 Rid

Observe | | 3 3 3 3 ≤ 3 | |3 | |3 | |3 x3 = x0 + x1 + x2 x0 + x1 + x2 .

230 on the smallest point on a diagonal cubic surface [Chap. VII √ √ 3 3 For (x0,...,x3) ∈ Rid, this implies | x3|≤ 3 | x2| and | x2|≥b0/ 3. We ﬁnd (1, ... ,1) CX ( Ê) 1 1 ∧ ∧ ωLeray = 2 dx0 dx1 dx2 3 x3 Rid Rid ≤ 1 1 ∧ ∧ 2 dx0 dx1 dx2 3 x2 Rid

b0 1 1 < dx dx dx 3 x2 2 1 0 √ 2 − | |∈ | | | |≥ 3 b0 x1 [ x0 ,b1] x2 b0/ 3 | |≥| | x2 x1 b0 ≤ 1 2√ 3 dx1 dx0 3 max{b0/ 3, | x1|} − | |∈ | | ⎡b0 x1 [ x0 ,b1] ⎤

b0 √ b0 ⎢ 3 ⎥ 2 ⎢ 3 1 ⎥ ≤ ⎣ dx1 dx0 + dx1 dx0⎦ 3 b0 | x1| −b b0 −b b0 0 | x1|∈[| x0|, √ ] 0 | x1|∈[ √ ,b1] 3 3 3 3

√ b0 √ 2 3 3 ≤ 2 · 4√b0 · 3 2 3b1 3 + 2log dx0 3 3 b0 3 b0 − √ b0 8 8 3 3b = b + b log 1 3 0 3 0 b 0 8 8 8 b1 = + log 3 b0 + b0 log . 3 9 3 b0

9.17. Corollary. For every ε>0, there exists a constant C such that − 1 1 (a0,... ,a3) ≤ ·| · · | 3 +ε · 3 τ∞ X ( Ê) C a0 ... a3 min ai ∞ i=0,... ,3 \{ } 4 for each (a0,...,a3) ∈ ( 0 ) .

Proof. We assume without restriction that | a0|≤ ... ≤|a3|. Then Lemma 9.15 and Proposition 9.16 together show that, for certain explicit positive constants C1 and C2, : ; | | (a ,... ,a ) − 1 1 1 3 a1 0 3 ≤| · · | 3 · | | 3 | | 3 τ∞ X ( Ê) a0 ... a3 C1 a0 + C2 a0 log | a0| − 1 1 1 | a1| = | a0 · ... · a3| 3 ·| a0| 3 C1 + C2 log 3 | a0|

1 1 − 3 ≤|a0 · ... · a3| 3 · min ai ∞ i=0,... ,3 1 · C + C log | a · ... · a | . 1 3 2 0 3

Sec. 9] the fundamental finiteness property 231

1 | · · |≤ | · · |ε There is a constant C such that C1 + 3 C2 log a0 ... a3 C a0 ... a3 for \{ } 4 every (a0,...,a3) ∈ ( 0 ) . iv. The Tamagawa number.

9.18. Proposition. For every ε>0, there exists a constant C>0 such that 1 1 H 1 : ... : 1 3 ≥ C · a0 a3 (a ,... ,a ) ε τ 0 3 |a0 · ... · a3| \{ } 4 for each (a0,...,a3) ∈ ( 0 ) . 2 Proof. We may assume that ε is small, say ε< 3 . Then, immediately from the deﬁnition of τ (a0,... ,a3),wehave

τ (a0,... ,a3) (a0,... ,a3) · (a0,... ,a3) · − t = α(X ) β(X ) lim (s 1) L s, χ (a0, ... ,a3) s→1 Pic(X ) É

· (a0,... ,a3) Br

τH X ( É ) ≤ (a0,... ,a3) · (a0,... ,a3) · − t α(X ) β(X ) lim (s 1) L s, χ (a0, ... ,a3) s→1 Pic(X ) É

· (a0,... ,a3)

τH X ( É ) (a0,... ,a3) · (a0,... ,a3) · − t = α(X ) β(X ) lim (s 1) L s, χ (a0, ... ,a3) s→1 Pic(X ) É (a0,... ,a3) · τν X ( Éν ) .

ν∈Val( É) Let us collect estimates for the factors. First, by Proposition 9.2, we have

ε − t ·| · · | 16 lim (s 1) L s, χ (a0, ... ,a3)

s→1 Pic(X ) É for a certain constant c1. Further, Proposition 9.13 yields 1 − ε 1 − ε (a0,... ,a3) ≤ ·| · · | 3 16 · 3 2 τp X ( Ép) c2 a0 ... a3 min ai p . i=0,... ,3 p prime p prime Finally, Corollary 9.17 shows − 1 ε 1 (a0,... ,a3) ≤ ·| · · | 3 + 2 · 3 τ∞ X ( Ê) C a0 ... a3 min ai ∞ . i=0,... ,3 We assert that the three inequalities together imply the following estimate for Peyre’s constant τ (a0,... ,a3) = τ(X(a0,... ,a3)), ε 1 (a0,... ,a3) 3 τ ≤ C3 ·|a0 · ... · a3| 2 · min ai p i=0,... ,3 p prime − ε 1 2 3 · min ai ∞ · min ai p . i=0,... ,3 i=0,... ,3 p prime

Indeed, this is trivial in the case τ (a0,... ,a3) =0.Otherwise,X(a0,... ,a3) has an adelic point and we may estimate the factors α and β by constants as shown in Section 3.

232 on the smallest point on a diagonal cubic surface [Chap. VII

By consequence,

− 1 − 1 3 3 min ai p · min ai ∞ 1 1 p prime i=0,... ,3 i=0,... ,3 ≥ · ε (a ,... ,a ) ε − τ 0 3 C 2 3 | a0 · ... · a3| 2 · min ai p p prime i=0,... ,3 1 1 3 3 1 · 1 max a max a i=0,... ,3 i p i=0,... ,3 i ∞ 1 · p prime = ε C3 ε (p) 2 | · · | 2 · a0 ... a3 max ai p prime i=0,... ,3 1 1 H 1 : ... : 1 3 · a0 a3 = ε . C3 ε (p) 2 | · · | 2 · a0 ... a3 max ai p prime i=0,... ,3 (p) ≤| (p) · · (p)| It is obvious that maxi=0,... ,3ai a0 ... a3 and | (p) · · (p)| | · · | a0 ... a3 = a0 ... a3 . p prime This shows

1 1 1 H 1 : ... : 1 3 ≥ · a0 a3 (a ,... ,a ) ε ε τ 0 3 C3 | a0 · ... · a3| 2 ·|a0 · ... · a3| 2 1 1 H 1 : ... : 1 3 · a0 a3 = ε . C3 | a0 · ... · a3|

3 9.19. Lemma. Let (a0 : ... : a3) ∈ P ( É) be any point such that a0 =0 ,...,a3 =0 .Then

1 1 3 H(a0 : ... : a3) ≤ H( : ... : ) . a0 a3 1 1 Proof. First, observe that (a0 : ... : a3) → : ... : is a well-de- a0 a3

ﬁned map. Hence, we may assume without restriction that a0, ... ,a3 ∈ and gcd(a0,...,a3)=1. This yields H(a0 : ... : a3)=maxi=0,... ,3|ai|. 1 1 On the other hand, ( : ... : )=(a1a2a3 : ... : a0a1a2).Consequently, a0 a3 1 1 3 3 H : ... : ≤ [max|ai|] = H(a0 : ... : a3) . a0 a3 i=0,... ,3 1 From this, the asserted inequality emerges when the roles of ai and are inter- ai changed.

9.20. Corollary. Let a0,...,a3 ∈ such that gcd(a0,...,a3)=1.Then 1 1 12 |a0 · ... · a3|≤H : ... : . a0 a3

4 4 Proof. Observe that | a0 · ... · a3|≤ max | ai| = H(a0 : ... : a3) and apply i=0,... ,3 Lemma 9.19.

Sec. 10] a negative result 233

9.21. Theorem. For each ε>0, there exists a constant C(ε) > 0 such that, \{ } 4 for all (a0,...,a3) ∈ ( 0 ) ,

1 1 −ε ≥ C(ε) · H 1 : ... : 1 3 . τ (a0,... ,a3) a0 a3

Proof. We may assume that gcd(a0,...,a3)=1. Then, by Proposition 9.18, 1 H 1 : ... : 1 3 1 a0 a3 ≥ C(ε) · ε . (a ,... ,a ) τ 0 3 | a · ... · a | 12 0 3 ε 1 1 ε Corollary 9.20 yields | a0 · ... · a3| 12 ≤ H : ... : . a0 a3

9.22. Corollary (Fundamental ﬁniteness). For each T>0, there are only ﬁnitely many diagonal cubic surfaces X(a0,... ,a3) : a x3 + ... + a x3 =0in P3 0 0 3 3 É such that τ (a0,... ,a3) >T. Proof. This is an immediate consequence of the comparison to the naive height established in Theorem 9.21.

10. A negative result

10.1. For an integer q =0 ,letX(q) ⊂ P3 be the cubic surface given by É qx3 +4y3 +2z3 + w3 =0,andlet

(q) (q) } m(X ):=min{ H(x:y :z :w) | (x:y :z :w) ∈ X ( É)

(q) (q) be the smallest height of a É-rational point on X . We compare m(X ) with the Tamagawa type number τ (q) := τ(X(q)).

10.2. Lemma. There is a constant C with the following property. For each pair (a, b) of natural numbers satisfying gcd(a, b)=1, there are two prime numbers 5.5 p1,p2 ≡ a (mod b), p1 = p2, such that p1,p2

10.3. Theorem. Assume the Generalized Riemann Hypothesis. Then there is no constant C such that C m(X(q)) < τ (q) \{ } for all q ∈ 0 .

Proof. We will construct a sequence (qi)i∈Æ of prime numbers such that (qi) (qi) qi ≡ 1(mod 72) and m(X ) τ →∞for i →∞. The proof will consist of several steps.

234 on the smallest point on a diagonal cubic surface [Chap. VII

First step. The prime qi divides exactly one of the four coeﬃcients in the equation deﬁning X(qi). In this case, it is known by the work of J.-L. Colliot-Thélène, D. Kanevsky, and J.-J. Sansuc [CT/K/S, Proof of Proposition 2] that precisely

Br 1

É τ X( É ) = τ X( ) . H 3 H It is therefore suﬃcient to verify that

(qi) (qi) (qi) t (qi)

· · · − · É →∞ m(X ) α(X ) β(X ) lim (s 1) L(s, χ (qi) ) τν X ( ν ) .

s→1 Pic(X ) É

ν∈Val( É) Second step. If q is a prime diﬀerent from 2,thenwehaverkPic(X(q))=1.

(q)

Ðº ÍÔÖ ÒÒ º½¾ É [Man, III, ]showsthatX is a minimal cubic surface over .

(q)

Ðº ÌÓÖÑ º½ By [Man, IV, ], we have Pic(X )= .

Third step. We have α(X(qi))=1and β(X(qi))=3.

α(X(qi))=1follows immediately from rk Pic(X(qi))=1. β(X(qi)) can be computed by the method indicated in Remark 3.4. We work with a group G ⊂ W (E6) of order 18, which decomposes the 27 lines into three orbits of nine lines each. An easy calculation shows

∩ ∼ Hom (NF F0)/NF0, É/ = /3 . (q) ≥ 3 q Fourth step. For the height of the smallest point, we have m(X ) 7 . 3 3 3 There are no rational solutions of the equation 4y +2z + w =0as this is impos- | |≥ | 3 3 3|≥ {| | | | | |} ≥ 3 q sible 2-adically. x 1 yields 4y +2z + w q and max y , z , w 7 . | |≥ (q) √1 Fifth step. For q 7, one has τ∞ X ( Ê) = I,whereI is independent of q. 3 |q| By Lemma 9.15, we have (1, ... ,1) (q) 1 CX ( Ê) τ∞(X )= ω . 4 3 |q| Leray (1, ... ,1)

√ CX √ ( Ê) √ |x |≤ 3 |q|,|x |≤ 3 4,|x |≤ 3 2,|x |≤1 √ √ 0 1 2 3 3 3 Since |x1|≤ 4, |x2|≤ 2, |x3|≤1,and | | 3 | 3 3 3|≤ 3 | |3 | |3 | 3| x0 = x1 + x2 + x3 x1 + x2 + x3 , 3 the condition |x0|≤ |q| is actually empty. (q) Sixth step. There is a positive constant C such that p primeτp X ( Ép) >Cfor every prime q ≡ 1(mod 72).

By Lemma 4.2, we have C1 > 0 such that (q) τp X ( Ép) >C1 . p prime p=2,3,q (q) It, therefore, remains to give lower bounds for the factors τ2 X ( É2) ,

(q) (q) É τ3 X ( É3) ,andτq X ( q) .

Sec. 10] a negative result 235

(q)

(q) 1 #X ( /8 )

É · As 2 q, by virtue of Sublemma 4.1, we have τ2 X ( 2) = 27 64 . Further,

(q)

≥ #X ( /8 ) 1

≡ − ∈ (q) since q 1(mod 8) implies (1:0:0:( 1)) X ( /8 ).

(q) (q) 2 7 · #X ( /9 ) ≡ Similarly, τ3 X ( É3) =(3 ) 81 . Again, q 1(mod 9) makes sure

− ∈ (q) (q) ≥ that (1:0:0:( 1)) X ( /9 ) and #X ( /9 ) 1. For the prime q, we argue a bit diﬀerently. First, −1 (q) Iq 7 7

det 1 − q Frobp | Pic(X ) ≥ (1 − 1/q) ≥ (72/73) . É

Furthermore, the reduction of X(q) modulo q is the cone over the elliptic curve given by 4y3 +2z3 + w3 =0. Therefore, on X(q) there are at least (q − 2 q +1)(q − 1)

smooth points deﬁned over q. As Hensel’s lemma may be applied to them, we get

(q) n − − #X ( /q ) (q 2 q +1)(q 1) 2 1 lim ≥ > 1 − 1 − n→∞ q2n q2 q q 72 2 ≥ 1 − √ . 73 73

≡

Seventh step. There is a sequence (qi)i∈Æ of primes such that qi 1(mod 72) and

[lim → (s − 1)L(s, χ (q ) )] →∞for i →∞. s 1 Pic(X i )

É (q )

(qi) i ⊗ 

Since rk Pic(X )=1, the representation Pic X contains exactly one trivial summand. Hence, É L s, χ (q ) = ζ(s) · L(s, χ (q ) )

Pic(X i ) P i É for a representation P (qi) not containing trivial components. Our goal is, therefore, →∞ →∞ to show L(1,χP (qi) ) for i .

Denote by Pi the i-th prime number p such that p ≡ 1(mod 3).Foreachi ∈ Æ, choose a cubic residue ri modulo Pi. Then deﬁne qi to be the second smallest prime number such that qi ≡ rj (mod Pj) for all j ∈{1,...,i} and qi ≡ 1(mod 72). This system of simultaneous congruences is solvable by the Chinese remainder theorem. Its solutions form an arithmetic progression with common diﬀer- ence 72P1 · ... · Pi. By Chebyshev’s inequalities, we know

θ(Pi) (2 log 2)Pi 72P1 · ... · Pi ≤ 72e < 72e .

According to our deﬁnition, qi is the second smallest prime in this arithmetic progression. From this, we clearly have that qi > 72P1 · ... · Pi →∞for i →∞. On the other hand, Lemma 10.2 shows

(2 log 2)Pi 5.5 (11 log 2)Pi qi ≤ C1 · (72e ) = C2e for certain constants C1 and C2.

236 on the smallest point on a diagonal cubic surface [Chap. VII

Corollary 5.9 gives us an estimate for the Artin conductor of the character χP (qi) . 12 6 12 6 6 (66 log 2)Pi We see Nχ ≤ 3 (a0 · ... · a3) =3 8 q ≤ C3 e for another con- P (qi) i stant C3.Consequently,

log Nχ ≤ (66 log 2)Pi +logC3 . P (qi)

1/2 We observe that (log Nχ ) ≤ Pi for i suﬃciently large. We assume from now P (qi) on that this inequality is fulﬁlled.

Recall that P (qi) is actually the direct sum of representations that are induced from one dimensional characters (Theorem 5.6). By consequence, it is known that · the Artin L-function L( ,χP (qi) ) is entire. Since we also assume the Generalized Riemann Hypothesis, we may apply the estimate of W. Duke [Duk, Proposition 5]. It shows −1 log L(1,χP (qi) )= χP (qi) (Frobp) p + O(1) . 1/2 p<(log Nχ ) P (qi) Here,

χ (q ) (Frob )=χ (q ) (Frob ) − 1 .

P i p Pic(X i ) p É ≡ For p 2(mod 3), this yields χP (qi) (Frobp)=0. On the other hand, for p ≡ 1(mod 3), we have, by virtue of Lemma 5.3,

2 2 2 χP (qi) (Frobp)=χ3(16q)+χ3(32q )+χ3(32q)+χ3(16q )+χ3(64q)+χ3(8q ) = χ (q)+χ (2q)+χ (4q)+χ (q2)+χ (2q2)+χ (4q2) 3 3 3 3 3 3 2 = 1+χ3(2) + χ3(4) χ3(q)+χ3(q ) .

This may be written down in an explicit form as ⎧ ⎪ ≡ ⎪ 0ifp 2(mod 3) , ⎨ ≡ 2 0ifp 1(mod 3) and p 3 =1, χ (q ) (Frobp)= q P i ⎪ 6ifp ≡ 1(mod 3), 2 =1, and i =1, ⎩⎪ p 3 p 3 − ≡ 2 qi 3ifp 1(mod 3), p 3 =1, and p 3 =1. 1/2 Modulo all primes p ≡ 1(mod 3), p<(log Nχ ) ≤ Pi,thenumberqi was (qi) P −1 constructed to be a cubic residue. Further, χP (qi) (Frob3)3 is of absolute value at most 2.Thus, 1 log L(1,χ (q ) )=6 + O(1) . P i p p≡1(mod 3) 2 ( p )3=1 1/2 p<(log Nχ ) P (qi) By the Chebotarev density theorem, the set of all primes, such that p ≡ 1(mod 3) 2 1 →∞ and p 3 =1,isofdensity 6 . We, therefore, have log L(1,χP (qi) ) as soon as we may guarantee Nχ →∞. P (qi) Since only a trivial character is are missing, we have, by Corollary 5.7,

| |1/2 ≥| |1/2 Nχ (q ) = Nχ (q ) = D(K1) D(K2) D(K3)/27 D(K3)/27 ,

P i Pic(X i ) É

Sec. 10] a negative result 237

3 3 É where, by choice of the coeﬃcients, K3 = É(ζ3, 64qi)= (ζ3, qi).Thereis the estimate

3 É | D(K3)| = Disc É(ζ3, qi)/

3 2 3 3

É · É É = Disc É( qi)/ N Disc (ζ3, qi)/ ( qi)

≥ 3 2 É Disc É( qi)/ .

Accordingto[Mc, Chapter 2, Exercise 41], we know

3 ≥ 2 É Disc É( qi)/ 3qi .

10.4. Remark. Note that the estimate for L(1,χP (qi) ) is the only point where we use the Generalized Riemann Hypothesis.

Appendix

Diophantus, with this name which is frequent in Greece, was a true Greek, disciple of Greek science, if one who towers high above his contemporaries. He was Greek in what he accomplished, as well as in what he was not able to accomplish. But we must not forget that Greek science, as it conquered the East from Alexandria ...,brought new ideas back home from these campaigns, that Greek mathematics as such has ceased to pick up whatever it found worth picking up here and there. Moritz Cantor (1907, translated by N. Schappacher)

1. A script in GAP

This is the script in GAP that was used to compute H1 G, Pic(X ) and rk Pic(X) É for all groups that may operate on the 27 lines upon a smooth cubic surface. The mathematical background is explained in Section 8, Subsection iv of Chap- ter III.

Table 1. A GAP script

# A script in GAP to compute H^1(G, Pic) = (NS \cap S_0) / NS_0 and the Picard rank for all # Galois groups of a cubic surface. # Here, S is the free abelian group on the 27 lines, S_0 \subset S the subgroup of all # principal divisors, and N: S --> S the norm map.

# We compute the intersection matrix of the 27 lines. # g must be a faithful representation of W(E_6) in S_27. SchnittMatrixBerechnen := function (g) local i, j, mat, len, tmp; mat := NullMat(27, 27); for i in [1..27] do for j in [1..27] do if (i = j) then mat[i][j] := -1; else tmp := [i, j]; Sort(tmp); len := Size(Orbit(g, tmp, OnSets)); # Operation of W(E_6) on unordered pairs of lines. # A pair of skew lines has orbit length 27*8. # A pair of intersecting lines has orbit length 27*5. if (len = 27*5) then mat[i][j] := 1; fi; fi; od; od; return mat; end;

# Compute the 27x27-matrix representing the norm map N: S --> S. NBerechnen := function (orb, u) local N, akt, j, k, l;

# The matrix N. N := NullMat(27, 27);

239

240 appendix

for j in [1..Size(orb)] do akt := orb[j]; for k in [1..Size(akt)] do for l in [1..Size(akt)] do N[akt[k]][akt[l]] := Size(u) / Size(akt); od; od; od; return N; end;

# Build up a matrix containing a minimal system of generators of NS in its columns. IBerechnen := function (orb, u) local my_I, j, k, akt;

# The matrix I. my_I := NullMat(27, Size(orb)); for j in [1..Size(orb)] do akt := orb[j]; for k in [1..Size(akt)] do # The line akt[k] lies in orbit akt numbered j. my_I[akt[k]][j] := Size(u) / Size(akt); od; od; return my_I; end;

# A separate routine in order to make a lattice base from the column vectors of a matrix. NormMat := function (m) return TransposedMat(HermiteNormalFormIntegerMat(BaseIntMat(TransposedMat(m)))); end;

# The discriminant of a lattice. The lattice needs not be maximal. # The lattice base is supposed to be given in the column vectors of m. # It is important that m is a lattice base. The function will return 0 for a linearly # dependent system. Disc := function (m) return Determinant(TransposedMat(m) * m); end;

# We compute H^1 of the Picard group and the arithmetic Picard rank using Manin’s formulas. h1_pic := function (schnitt_matrix, u) local orb, I_mat, AI, N, K, linke_seite, rechte_seite, disc_links, disc_rechts, index, rang, mul, scal_links, schnitt;

# Everything depends only on the combinatorial structure of the orbits. orb := Orbits(u, [1..27]);

I_mat := IBerechnen(orb, u); # I_mat represents NS. AI := schnitt_matrix * I_mat; N := NBerechnen(orb, u); # N represents the norm map.

# Normalization: We choose all integral vectors in the kernel. # Computation of NS \cap S_0. K := NullspaceIntMat(TransposedMat(AI)); # K collects the relations among the generators of NS modulo S_0. linke_seite := NormMat(I_mat * TransposedMat(K)); # linke_seite represents the lattice NS \cap S_0.

K := NullspaceIntMat(TransposedMat(schnitt_matrix)); # The rows of K are generators of S_0. rechte_seite := NormMat(N * TransposedMat(K)); # The columns of N * TransposedMat(K) are generators of NS_0.

# The discriminats. disc_links := Disc(linke_seite); disc_rechts := Disc(rechte_seite);

index := RootInt(disc_rechts / disc_links); # The order of H^1(G, Pic). rang := Size(orb) - Size(TransposedMat(rechte_seite)); # The arithmetic Picard rank.

AppendTo("h1pic.txt"," #H^1 = ", index); if index > 3 then # We have to compute the primary decomposition. # This code suffices since index may be only 1, 2, 3, 4, or 9. if index = 4 then mul := 2; else mul := 3; fi; scal_links := mul * linke_seite; schnitt := TransposedMat(BaseIntersectionIntMats(TransposedMat(rechte_seite), TransposedMat(scal_links))); if Disc(schnitt) = Disc(scal_links) then AppendTo("h1pic.txt"," [ ", mul, ", ", mul, " ]");

appendix 241

else AppendTo("h1pic.txt"," More complicated than the direct sum of Z/", mul, "Z"); fi; fi; AppendTo("h1pic.txt", ", Rk(Pic) = ", rang, ","); end;

# Adds the orbit structure of the lines to the list. bahnstruktur := function (u) local ol;

ol := ShallowCopy(OrbitLengths(u, [1..27])); Sort(ol); AppendTo("h1pic.txt", " Orbits ", ol); end;

# Our group taken from the library. we6 := TransitiveGroup(27, 1161); schnitt_matrix := SchnittMatrixBerechnen(we6); # The subgroups. ugv := ConjugacyClassesSubgroups(we6);;

# We compute H^1(G, Pic) for all Galois groups. for i in [1..Size(ugv)] do u := Representative(ugv[i]); AppendTo("h1pic.txt", i, " #U = ", Size(u), " "); AppendTo("h1pic.txt", AbelianInvariants(u), ",");

if IsTransitive(u) and (Size(u) > 1) then AppendTo("h1pic.txt", " transitive"); else h1_pic(schnitt_matrix, u); bahnstruktur(u);

fi; AppendTo("h1pic.txt", "\n"); od;

2. The list

This is the list produced by the GAP script. For each subgroup ofW (E6),wegive

its abelian quotient, the corresponding values of H1 G, Pic(X ) and rk Pic(X), É and the orbit structure of the 27 lines. We make use of this list in several ways. For our presentation of the Brauer– Manin obstruction in the case of diagonal cubic surfaces given in Section IV.6, the list is of fundamental importance. On the other hand, we also use it in the arguments given in IV.5.33 and Remarks IV.5.37. It is further applied in Examples II.5.8 and II.5.9. Here, the goal is simply to show that one has Picard rank 2 in the situations considered. Finally, we give reference to the list in Remarks II.6.2.ii) in order to prove that the factor β is bounded from above by 9.

Table 2. H1(G, Pic) and rk Pic(X) for smooth cubic surfaces

1#U=1[],#H^1 = 1, Rk(Pic) = 7, Orbits [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ] 2#U=2[2],#H^1 = 1, Rk(Pic) = 6, Orbits [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2 ] 3#U=2[2],#H^1 = 4 [ 2, 2 ], Rk(Pic) = 3, Orbits [ 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 ] 4#U=2[2],#H^1 = 1, Rk(Pic) = 5, Orbits [ 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 ] 5#U=2[2],#H^1 = 1, Rk(Pic) = 4, Orbits [ 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,2] 6#U=3[3],#H^1 = 9 [ 3, 3 ], Rk(Pic) = 1, Orbits [ 3, 3, 3, 3, 3, 3, 3, 3, 3 ] 7#U=3[3],#H^1 = 1, Rk(Pic) = 5, Orbits [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3, 3, 3, 3,3] 8#U=3[3],#H^1 = 1, Rk(Pic) = 3, Orbits [ 3, 3, 3, 3, 3, 3, 3, 3, 3 ]

242 appendix

9#U=4[4],#H^1 = 1, Rk(Pic) = 3, Orbits [ 1, 1, 1, 4, 4, 4, 4, 4, 4 ] 10#U=4[2,2],#H^1 = 1, Rk(Pic) = 5, Orbits [ 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 4 ] 11#U=4[2,2],#H^1 = 1, Rk(Pic) = 4, Orbits [ 1, 1, 1, 1, 1, 2, 2, 2, 4, 4, 4, 4 ] 12#U=4[2,2],#H^1 =4[2,2],Rk(Pic) = 2, Orbits [ 1, 2, 2, 2, 2, 2, 4, 4, 4, 4 ] 13#U=4[4],#H^1 = 4 [ 2, 2 ], Rk(Pic) = 2, Orbits [ 1, 2, 2, 2, 4, 4, 4, 4, 4 ] 14#U=4[2,2],#H^1 = 2, Rk(Pic) = 3, Orbits [ 1, 1, 1, 2, 2, 2, 2, 4, 4, 4, 4 ] 15#U=4[2,2],#H^1 = 1, Rk(Pic) = 4, Orbits [ 1, 1, 1, 2, 2, 2, 2, 2, 2, 4, 4, 4 ] 16#U=4[2,2],#H^1 = 2, Rk(Pic) = 3, Orbits [ 1, 1, 1, 2, 2, 2, 2, 2, 2, 4, 4, 4 ] 17#U=4[2,2],#H^1 = 2, Rk(Pic) = 3, Orbits [ 1, 1, 1, 2, 2, 4, 4, 4, 4, 4 ] 18#U=4[4],#H^1 = 1, Rk(Pic) = 4, Orbits [ 1, 1, 1, 1, 1, 2, 4, 4, 4, 4, 4 ] 19#U=4[2,2],#H^1 = 1, Rk(Pic) = 3, Orbits [ 1, 2, 2, 2, 2, 2, 4, 4, 4, 4 ] 20#U=4[4],#H^1 = 1, Rk(Pic) = 3, Orbits [ 1, 2, 2, 2, 4, 4, 4, 4, 4 ] 21#U=4[2,2],#H^1 = 2, Rk(Pic) = 2, Orbits [ 1, 2, 2, 2, 4, 4, 4, 4, 4 ] 22#U=4[2,2],#H^1 = 1, Rk(Pic) = 4, Orbits [ 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 4, 4 ] 23#U=5[5],#H^1 = 1, Rk(Pic) = 3, Orbits [ 1, 1, 5, 5, 5, 5, 5 ] 24#U=6[2],#H^1 = 1, Rk(Pic) = 5, Orbits [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3, 3, 3, 3,3] 25#U=6[2],#H^1 = 4 [ 2, 2 ], Rk(Pic) = 1, Orbits [ 3, 3, 3, 6, 6, 6 ] 26#U=6[2,3],#H^1 = 1, Rk(Pic) = 1, Orbits [ 3, 6, 6, 6, 6 ] 27#U=6[2,3],#H^1 =4[2,2],Rk(Pic) = 1, Orbits [ 3, 6, 6, 6, 6 ] 28#U=6[2,3],#H^1 = 1, Rk(Pic) = 4, Orbits [ 1, 1, 1, 2, 2, 2, 3, 3, 3, 3, 6 ] 29#U=6[2],#H^1 = 1, Rk(Pic) = 2, Orbits [ 3, 3, 3, 6, 6, 6 ] 30#U=6[2,3],#H^1 = 1, Rk(Pic) = 3, Orbits [ 1, 1, 1, 2, 2, 2, 6, 6, 6 ] 31#U=6[2],#H^1 = 1, Rk(Pic) = 3, Orbits [ 1, 1, 1, 2, 2, 2, 6, 6, 6 ] 32#U=6[2,3],#H^1 = 1, Rk(Pic) = 2, Orbits [ 3, 3, 3, 3, 3, 6, 6 ] 33#U=6[2],#H^1 = 1, Rk(Pic) = 4, Orbits [ 1, 1, 1, 2, 2, 2, 3, 3, 3, 3, 6 ] 34#U=6[2],#H^1 = 1, Rk(Pic) = 3, Orbits [ 3, 3, 3, 3, 3, 3, 3, 6 ] 35#U=6[2],#H^1 = 1, Rk(Pic) = 3, Orbits [ 1, 2, 2, 2, 2, 3, 3, 6, 6 ] 36#U=6[2,3],#H^1 = 1, Rk(Pic) = 3, Orbits [ 1, 2, 2, 2, 2, 3, 3, 6, 6 ] 37#U=6[2],#H^1 = 3, Rk(Pic) = 1, Orbits [ 3, 3, 3, 6, 6, 6 ] 38#U=6[2],#H^1 = 1, Rk(Pic) = 2, Orbits [ 3, 3, 3, 6, 6, 6 ] 39#U=6[2,3],#H^1 = 1, Rk(Pic) = 2, Orbits [ 3, 6, 6, 6, 6 ] 40#U=8[2,2],#H^1 = 1, Rk(Pic) = 3, Orbits [ 1, 1, 1, 8, 8, 8 ] 41#U=8[2,2,2],#H^1 = 1, Rk(Pic) = 3, Orbits [ 1, 1, 1, 2, 2, 2, 2, 8, 8 ] 42#U=8[2,2,2],#H^1 = 2, Rk(Pic) = 3, Orbits [ 1, 1, 1, 4, 4, 4, 4, 4, 4 ] 43#U=8[2,2,2],#H^1 = 2, Rk(Pic) = 2, Orbits [ 1, 2, 2, 2, 2, 2, 8, 8 ] 44#U=8[2,2],#H^1 =4[2,2],Rk(Pic) = 2, Orbits [ 1, 2, 2, 2, 4, 4, 4, 4, 4 ] 45#U=8[2,4],#H^1 = 1, Rk(Pic) = 3, Orbits [ 1, 1, 1, 4, 4, 8, 8 ] 46#U=8[2,2,2],#H^1 = 1, Rk(Pic) = 4, Orbits [ 1, 1, 1, 2, 2, 2, 2, 2, 2, 4, 4, 4 ] 47#U=8[2,2,2],#H^1 = 1, Rk(Pic) = 3, Orbits [ 1, 2, 2, 2, 2, 2, 4, 4, 8 ] 48#U=8[2,2,2],#H^1 = 2, Rk(Pic) = 3, Orbits [ 1, 1, 1, 4, 4, 4, 4, 4, 4 ] 49#U=8[2,2],#H^1 = 1, Rk(Pic) = 4, Orbits [ 1, 1, 1, 1, 1, 2, 4, 4, 4, 4, 4 ] 50#U=8[2,2],#H^1 = 2, Rk(Pic) = 2, Orbits [ 1, 2, 2, 2, 4, 8, 8 ] 51#U=8[2,2],#H^1 = 2, Rk(Pic) = 2, Orbits [ 1, 2, 2, 2, 4, 8, 8 ] 52#U=8[2,4],#H^1 = 2, Rk(Pic) = 2, Orbits [ 1, 2, 4, 4, 8, 8 ] 53#U=8[2,2,2],#H^1 = 2, Rk(Pic) = 3, Orbits [ 1, 1, 1, 2, 2, 4, 4, 4, 4, 4 ] 54#U=8[2,2],#H^1 = 1, Rk(Pic) = 3, Orbits [ 1, 1, 1, 4, 4, 4, 4, 8 ] 55#U=8[2,2,2],#H^1 = 2, Rk(Pic) = 2, Orbits [ 1, 2, 2, 2, 4, 8, 8 ] 56#U=8[2,4],#H^1 = 2, Rk(Pic) = 2, Orbits [ 1, 2, 2, 2, 4, 8, 8 ] 57#U=8[2,4],#H^1 = 1, Rk(Pic) = 3, Orbits [ 1, 1, 1, 2, 2, 4, 8, 8 ] 58#U=8[2,2],#H^1 = 1, Rk(Pic) = 3, Orbits [ 1, 2, 2, 2, 4, 4, 4, 8 ] 59#U=8[2,2],#H^1 = 1, Rk(Pic) = 3, Orbits [ 1, 2, 2, 2, 4, 4, 4, 8 ] 60#U=8[2,4],#H^1 = 2, Rk(Pic) = 2, Orbits [ 1, 2, 4, 4, 4, 4, 8 ] 61#U=8[2,2],#H^1 = 1, Rk(Pic) = 3, Orbits [ 1, 2, 2, 2, 4, 4, 4, 4, 4 ] 62#U=8[2,2],#H^1 = 2, Rk(Pic) = 2, Orbits [ 1, 2, 4, 4, 8, 8 ] 63#U=8[2,2],#H^1 = 1, Rk(Pic) = 3, Orbits [ 1, 1, 1, 2, 2, 4, 8, 8 ] 64#U=8[2,2],#H^1 = 2, Rk(Pic) = 2, Orbits [ 1, 2, 4, 4, 4, 4, 8 ] 65#U=8[2,2],#H^1 = 2, Rk(Pic) = 2, Orbits [ 1, 2, 4, 4, 4, 4, 8 ] 66#U=8[2,2],#H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 2, 4, 4, 8, 8 ] 67#U=8[8],#H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 2, 8, 8, 8 ] 68#U=8[2,4],#H^1 = 1, Rk(Pic) = 3, Orbits [ 1, 2, 2, 2, 4, 4, 4, 8 ] 69#U=8[2,2,2],#H^1 = 2, Rk(Pic) = 2, Orbits [ 1, 2, 2, 2, 4, 4, 4, 8 ] 70#U=9[3,3],#H^1 = 1, Rk(Pic) = 3, Orbits [ 3, 3, 3, 3, 3, 3, 9 ] 71#U=9[3,3],#H^1 = 3, Rk(Pic) = 1, Orbits [ 9, 9, 9 ] 72#U=9[3,3],#H^1 = 3, Rk(Pic) = 1, Orbits [ 3, 3, 3, 9, 9 ] 73#U=9[9],#H^1 = 1, Rk(Pic) = 1, Orbits [ 9, 9, 9 ] 74 #U = 10 [ 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 2, 5, 5, 5, 10 ] 75 #U = 10 [ 2, 5 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 2, 5, 5, 5, 10 ] 76 #U = 10 [ 2 ], #H^1 = 1, Rk(Pic) = 3, Orbits [ 1, 1, 5, 5, 5, 5, 5 ] 77 #U = 12 [ 3 ], #H^1 = 1, Rk(Pic) = 4, Orbits [ 1, 1, 1, 1, 1, 4, 4, 4, 4, 6 ] 78 #U = 12 [ 3 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 3, 6, 6, 12 ] 79 #U = 12 [ 2, 2 ], #H^1 = 1, Rk(Pic) = 4, Orbits [ 1, 1, 1, 2, 2, 2, 3, 3, 3, 3, 6 ] 80 #U = 12 [ 2, 2 ], #H^1 = 1, Rk(Pic) = 3, Orbits [ 1, 1, 1, 2, 2, 2, 6, 6, 6 ] 81 #U = 12 [ 2, 2 ], #H^1 = 2, Rk(Pic) = 1, Orbits [ 3, 3, 3, 6, 12 ] 82 #U = 12 [ 2, 2 ], #H^1 =4[2,2],Rk(Pic) = 1, Orbits [ 3, 6, 6, 6, 6 ] 83 #U = 12 [ 2, 2, 3 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 2, 2, 4, 6, 12 ] 84 #U = 12 [ 3, 4 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 4, 4, 6, 12 ] 85 #U = 12 [ 2, 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 2, 2, 4, 6, 12 ] 86 #U = 12 [ 4 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 4, 4, 6, 12 ] 87 #U = 12 [ 2, 2 ], #H^1 = 1, Rk(Pic) = 3, Orbits [ 1, 2, 2, 3, 3, 4, 6, 6 ] 88 #U = 12 [ 2, 2, 3 ], #H^1 = 1, Rk(Pic) = 3, Orbits [ 1, 2, 2, 3, 3, 4, 6, 6 ] 89 #U = 12 [ 2, 2 ], #H^1 = 1, Rk(Pic) = 3, Orbits [ 1, 2, 2, 2, 2, 3, 3, 6, 6 ]

appendix 243

90 #U = 12 [ 3, 4 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 3, 12, 12 ] 91 #U = 12 [ 2, 2 ], #H^1 = 2, Rk(Pic) = 1, Orbits [ 3, 6, 6, 12 ] 92 #U = 12 [ 2, 2 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 3, 6, 6, 12 ] 93 #U = 12 [ 2, 2, 3 ], #H^1 = 2, Rk(Pic) = 1, Orbits [ 3, 6, 6, 12 ] 94 #U = 12 [ 2, 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 3, 3, 3, 6, 6, 6 ] 95 #U = 12 [ 2, 2 ], #H^1 = 2, Rk(Pic) = 1, Orbits [ 3, 6, 6, 12 ] 96 #U = 12 [ 2, 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 2, 2, 4, 6, 12 ] 97 #U = 12 [ 2, 2 ], #H^1 = 1, Rk(Pic) = 3, Orbits [ 1, 2, 2, 3, 3, 4, 6, 6 ] 98 #U = 12 [ 2, 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 3, 6, 6, 6, 6 ] 99 #U = 16 [ 2, 2, 2, 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 2, 2, 2, 2, 2, 16 ] 100 #U = 16 [ 2, 2, 2, 2 ], #H^1 = 2, Rk(Pic) = 3, Orbits [ 1, 1, 1, 4, 4, 4, 4, 4, 4 ] 101 #U = 16 [ 2, 2, 4 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 2, 2, 2, 4, 16 ] 102 #U = 16 [ 2, 2, 2 ], #H^1 = 1, Rk(Pic) = 3, Orbits [ 1, 1, 1, 8, 8, 8 ] 103 #U = 16 [ 2, 2, 2, 2 ], #H^1 = 2, Rk(Pic) = 2, Orbits [ 1, 2, 2, 2, 4, 8, 8 ] 104 #U = 16 [ 2, 2, 2 ], #H^1 = 1, Rk(Pic) = 3, Orbits [ 1, 1, 1, 4, 4, 8, 8 ] 105 #U = 16 [ 2, 4 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 2, 4, 4, 16 ] 106 #U = 16 [ 2, 4 ], #H^1 = 1, Rk(Pic) = 3, Orbits [ 1, 1, 1, 4, 4, 8, 8 ] 107 #U = 16 [ 2, 2, 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 2, 4, 4, 16 ] 108 #U = 16 [ 4, 4 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 2, 4, 4, 16 ] 109 #U = 16 [ 2, 2, 2 ], #H^1 = 2, Rk(Pic) = 2, Orbits [ 1, 2, 4, 4, 8, 8 ] 110 #U = 16 [ 2, 2, 2 ], #H^1 = 1, Rk(Pic) = 3, Orbits [ 1, 1, 1, 2, 2, 4, 8, 8 ] 111 #U = 16 [ 2, 4 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 2, 4, 4, 16 ] 112 #U = 16 [ 2, 2, 2 ], #H^1 = 2, Rk(Pic) = 2, Orbits [ 1, 2, 4, 4, 8, 8 ] 113 #U = 16 [ 2, 4 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 2, 4, 4, 16 ] 114 #U = 16 [ 2, 2, 2 ], #H^1 = 2, Rk(Pic) = 2, Orbits [ 1, 2, 4, 4, 8, 8 ] 115 #U = 16 [ 2, 4 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 2, 4, 4, 16 ] 116 #U = 16 [ 2, 2, 4 ], #H^1 = 2, Rk(Pic) = 2, Orbits [ 1, 2, 4, 4, 8, 8 ] 117 #U = 16 [ 2, 2, 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 2, 2, 2, 4, 16 ] 118 #U = 16 [ 2, 4 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 2, 8, 16 ] 119 #U = 16 [ 2, 2, 2 ], #H^1 = 2, Rk(Pic) = 2, Orbits [ 1, 2, 2, 2, 4, 8, 8 ] 120 #U = 16 [ 2, 2, 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 2, 2, 2, 4, 16 ] 121 #U = 16 [ 2, 4 ], #H^1 = 1, Rk(Pic) = 3, Orbits [ 1, 1, 1, 4, 4, 8, 8 ] 122 #U = 16 [ 2, 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 2, 8, 16 ] 123 #U = 16 [ 2, 2, 2 ], #H^1 = 2, Rk(Pic) = 2, Orbits [ 1, 2, 4, 4, 4, 4, 8 ] 124 #U = 16 [ 2, 2, 2 ], #H^1 = 2, Rk(Pic) = 2, Orbits [ 1, 2, 4, 4, 8, 8 ] 125 #U = 16 [ 2, 2, 2 ], #H^1 = 1, Rk(Pic) = 3, Orbits [ 1, 2, 2, 2, 4, 4, 4, 8 ] 126 #U = 16 [ 2, 2, 2 ], #H^1 = 2, Rk(Pic) = 2, Orbits [ 1, 2, 4, 4, 8, 8 ] 127 #U = 16 [ 2, 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 2, 8, 8, 8 ] 128 #U = 18 [ 2 ], #H^1 = 1, Rk(Pic) = 3, Orbits [ 3, 3, 3, 3, 3, 3, 9 ] 129 #U = 18 [ 2, 3 ], #H^1 = 3, Rk(Pic) = 1, Orbits [ 3, 3, 3, 9, 9 ] 130 #U = 18 [ 2, 3 ], #H^1 = 1, Rk(Pic) = 3, Orbits [ 3, 3, 3, 3, 3, 3, 9 ] 131 #U = 18 [ 2, 3 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 3, 3, 3, 18 ] 132 #U = 18 [ 2 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 3, 3, 3, 18 ] 133 #U = 18 [ 2, 3 ], #H^1 =4[2,2],Rk(Pic) = 1, Orbits [ 6, 6, 6, 9 ] 134 #U = 18 [ 2, 3, 3 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 3, 3, 6, 6, 9 ] 135 #U = 18 [ 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 3, 3, 6, 6, 9 ] 136 #U = 18 [ 2, 3 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 9, 18 ] 137 #U = 18 [ 2 ], #H^1 = 3, Rk(Pic) = 1, Orbits [ 9, 9, 9 ] 138 #U = 18 [ 2 ], #H^1 = 3, Rk(Pic) = 1, Orbits [ 3, 6, 9, 9 ] 139 #U = 18 [ 2, 3 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 3, 6, 18 ] 140 #U = 18 [ 2, 3 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 3, 3, 6, 6, 9 ] 141 #U = 18 [ 2, 3 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 6, 6, 6, 9 ] 142 #U = 18 [ 2, 3, 3 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 3, 6, 18 ] 143 #U = 18 [ 2, 3 ], #H^1 = 3, Rk(Pic) = 1, Orbits [ 3, 6, 9, 9 ] 144 #U = 18 [ 2 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 9, 9, 9 ] 145 #U = 18 [ 2, 3 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 9, 18 ] 146 #U = 20 [ 4 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 2, 5, 10, 10 ] 147 #U = 20 [ 2, 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 2, 5, 5, 5, 10 ] 148 #U = 20 [ 4 ], #H^1 = 1, Rk(Pic) = 3, Orbits [ 1, 1, 5, 5, 5, 10 ] 149 #U = 24 [ 3 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 3, 24 ] 150 #U = 24 [ 2 ], #H^1 = 2, Rk(Pic) = 2, Orbits [ 1, 2, 2, 6, 8, 8 ] 151 #U = 24 [ 2, 3 ], #H^1 = 2, Rk(Pic) = 2, Orbits [ 1, 2, 2, 6, 8, 8 ] 152 #U = 24 [ 2 ], #H^1 = 1, Rk(Pic) = 4, Orbits [ 1, 1, 1, 1, 1, 4, 4, 4, 4, 6 ] 153 #U = 24 [ 2, 2, 3 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 3, 24 ] 154 #U = 24 [ 3 ], #H^1 = 1, Rk(Pic) = 3, Orbits [ 1, 1, 1, 8, 8, 8 ] 155 #U = 24 [ 3 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 3, 24 ] 156 #U = 24 [ 2 ], #H^1 = 1, Rk(Pic) = 3, Orbits [ 1, 2, 2, 4, 4, 6, 8 ] 157 #U = 24 [ 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 3, 6, 6, 12 ] 158 #U = 24 [ 2 ], #H^1 = 2, Rk(Pic) = 1, Orbits [ 3, 12, 12 ] 159 #U = 24 [ 2, 3 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 3, 6, 6, 12 ] 160 #U = 24 [ 2 ], #H^1 = 1, Rk(Pic) = 3, Orbits [ 1, 1, 1, 2, 6, 8, 8 ] 161 #U = 24 [ 2, 3 ], #H^1 = 2, Rk(Pic) = 1, Orbits [ 3, 12, 12 ] 162 #U = 24 [ 2, 3 ], #H^1 = 2, Rk(Pic) = 1, Orbits [ 3, 12, 12 ] 163 #U = 24 [ 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 3, 6, 6, 12 ] 164 #U = 24 [ 2 ], #H^1 = 2, Rk(Pic) = 1, Orbits [ 3, 12, 12 ] 165 #U = 24 [ 2, 3 ], #H^1 = 1, Rk(Pic) = 3, Orbits [ 1, 2, 2, 4, 4, 6, 8 ] 166 #U = 24 [ 2, 3 ], #H^1 = 1, Rk(Pic) = 3, Orbits [ 1, 1, 1, 2, 6, 8, 8 ] 167 #U = 24 [ 2, 2, 3 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 4, 4, 6, 12 ] 168 #U = 24 [ 2, 2, 2 ], #H^1 = 1, Rk(Pic) = 3, Orbits [ 1, 2, 2, 3, 3, 4, 6, 6 ] 169 #U = 24 [ 8 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 3, 24 ] 170 #U = 24 [ 2, 2, 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 2, 2, 4, 6, 12 ] 171 #U = 24 [ 2, 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 4, 4, 6, 12 ] 172 #U = 24 [ 2, 2 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 3, 12, 12 ]

244 appendix

173 #U = 24 [ 2, 4 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 4, 4, 6, 12 ] 174 #U = 24 [ 2, 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 4, 4, 6, 12 ] 175 #U = 24 [ 2, 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 4, 4, 6, 12 ] 176 #U = 24 [ 2, 2, 2 ], #H^1 = 2, Rk(Pic) = 1, Orbits [ 3, 6, 6, 12 ] 177 #U = 27 [ 3, 3 ], transitive 178 #U = 27 [ 3, 3, 3 ], #H^1 = 3, Rk(Pic) = 1, Orbits [ 9, 9, 9 ] 179 #U = 27 [ 3, 3 ], transitive 180 #U = 32 [ 2, 2, 2, 2 ], #H^1 = 1, Rk(Pic) = 3, Orbits [ 1, 1, 1, 8, 8, 8 ] 181 #U = 32 [ 2, 2, 2, 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 2, 2, 2, 4, 16 ] 182 #U = 32 [ 2, 4 ], #H^1 = 1, Rk(Pic) = 3, Orbits [ 1, 1, 1, 8, 8, 8 ] 183 #U = 32 [ 2, 2, 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 2, 4, 4, 16 ] 184 #U = 32 [ 2, 4 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 2, 8, 16 ] 185 #U = 32 [ 2, 2, 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 2, 4, 4, 16 ] 186 #U = 32 [ 2, 4 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 2, 8, 16 ] 187 #U = 32 [ 2, 2, 2 ], #H^1 = 1, Rk(Pic) = 3, Orbits [ 1, 1, 1, 4, 4, 8, 8 ] 188 #U = 32 [ 2, 2, 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 2, 4, 4, 16 ] 189 #U = 32 [ 2, 2, 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 2, 4, 4, 16 ] 190 #U = 32 [ 2, 4 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 2, 8, 16 ] 191 #U = 32 [ 2, 2, 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 2, 8, 16 ] 192 #U = 32 [ 2, 2, 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 2, 4, 4, 16 ] 193 #U = 32 [ 2, 2, 2, 2 ], #H^1 = 2, Rk(Pic) = 2, Orbits [ 1, 2, 4, 4, 8, 8 ] 194 #U = 32 [ 2, 2, 4 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 2, 4, 4, 16 ] 195 #U = 36 [ 4 ], #H^1 =4[2,2],Rk(Pic) = 1, Orbits [ 6, 6, 6, 9 ] 196 #U = 36 [ 2, 2 ], #H^1 =4[2,2],Rk(Pic) = 1, Orbits [ 6, 6, 6, 9 ] 197 #U = 36 [ 2, 2 ], #H^1 = 1, Rk(Pic) = 3, Orbits [ 3, 3, 3, 3, 3, 3, 9 ] 198 #U = 36 [ 2, 2 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 3, 3, 3, 18 ] 199 #U = 36 [ 2, 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 3, 3, 6, 6, 9 ] 200 #U = 36 [ 2, 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 3, 3, 6, 6, 9 ] 201 #U = 36 [ 4 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 6, 6, 6, 9 ] 202 #U = 36 [ 2, 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 6, 6, 6, 9 ] 203 #U = 36 [ 2, 2 ], #H^1 = 2, Rk(Pic) = 1, Orbits [ 6, 9, 12 ] 204 #U = 36 [ 2, 2 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 3, 6, 18 ] 205 #U = 36 [ 2, 2 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 3, 6, 18 ] 206 #U = 36 [ 2, 2, 3 ], #H^1 = 2, Rk(Pic) = 1, Orbits [ 6, 9, 12 ] 207 #U = 36 [ 2, 2 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 3, 6, 18 ] 208 #U = 36 [ 2, 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 3, 3, 6, 6, 9 ] 209 #U = 36 [ 2, 2, 3 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 3, 6, 18 ] 210 #U = 36 [ 2, 2, 3 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 3, 6, 18 ] 211 #U = 36 [ 2, 2 ], #H^1 = 3, Rk(Pic) = 1, Orbits [ 3, 6, 9, 9 ] 212 #U = 36 [ 2, 2, 3 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 3, 3, 6, 6, 9 ] 213 #U = 36 [ 2, 2 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 9, 18 ] 214 #U = 40 [ 2, 4 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 2, 5, 10, 10 ] 215 #U = 48 [ 2, 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 2, 2, 6, 16 ] 216 #U = 48 [ 2, 3 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 3, 24 ] 217 #U = 48 [ 4 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 4, 6, 16 ] 218 #U = 48 [ 2, 2 ], #H^1 = 2, Rk(Pic) = 2, Orbits [ 1, 2, 2, 6, 8, 8 ] 219 #U = 48 [ 2, 2, 3 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 2, 2, 6, 16 ] 220 #U = 48 [ 3, 4 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 4, 6, 16 ] 221 #U = 48 [ 2, 2, 3 ], #H^1 = 2, Rk(Pic) = 2, Orbits [ 1, 4, 6, 8, 8 ] 222 #U = 48 [ 2, 2 ], #H^1 = 2, Rk(Pic) = 2, Orbits [ 1, 4, 6, 8, 8 ] 223 #U = 48 [ 2, 2 ], #H^1 = 2, Rk(Pic) = 1, Orbits [ 3, 12, 12 ] 224 #U = 48 [ 2, 2 ], #H^1 = 1, Rk(Pic) = 3, Orbits [ 1, 2, 2, 4, 4, 6, 8 ] 225 #U = 48 [ 2, 2 ], #H^1 = 1, Rk(Pic) = 3, Orbits [ 1, 1, 1, 2, 6, 8, 8 ] 226 #U = 48 [ 2, 2, 3 ], #H^1 = 2, Rk(Pic) = 1, Orbits [ 3, 12, 12 ] 227 #U = 48 [ 2, 2 ], #H^1 = 2, Rk(Pic) = 1, Orbits [ 3, 12, 12 ] 228 #U = 48 [ 2, 2 ], #H^1 = 2, Rk(Pic) = 1, Orbits [ 3, 12, 12 ] 229 #U = 48 [ 2, 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 3, 6, 6, 12 ] 230 #U = 48 [ 2, 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 2, 2, 6, 16 ] 231 #U = 48 [ 2, 2 ], #H^1 = 2, Rk(Pic) = 2, Orbits [ 1, 4, 6, 8, 8 ] 232 #U = 48 [ 2 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 3, 24 ] 233 #U = 48 [ 2, 2 ], #H^1 = 2, Rk(Pic) = 1, Orbits [ 3, 12, 12 ] 234 #U = 48 [ 2, 2 ], #H^1 = 2, Rk(Pic) = 1, Orbits [ 3, 12, 12 ] 235 #U = 48 [ 2, 2 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 3, 24 ] 236 #U = 48 [ 2 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 3, 24 ] 237 #U = 48 [ 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 2, 8, 16 ] 238 #U = 48 [ 2, 2, 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 4, 4, 6, 12 ] 239 #U = 54 [ 2 ], transitive 240 #U = 54 [ 2 ], #H^1 = 3, Rk(Pic) = 1, Orbits [ 9, 9, 9 ] 241 #U = 54 [ 2, 3 ], #H^1 = 3, Rk(Pic) = 1, Orbits [ 9, 9, 9 ] 242 #U = 54 [ 2, 3, 3 ], #H^1 = 3, Rk(Pic) = 1, Orbits [ 9, 9, 9 ] 243 #U = 54 [ 2, 3 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 9, 18 ] 244 #U = 54 [ 2, 3, 3 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 9, 18 ] 245 #U = 54 [ 2, 3 ], transitive 246 #U = 54 [ 2, 3 ], transitive 247 #U = 60 [ ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 6, 6, 15 ] 248 #U = 60 [ ], #H^1 = 1, Rk(Pic) = 3, Orbits [ 1, 1, 5, 5, 5, 10 ] 249 #U = 64 [ 2, 2, 2 ], #H^1 = 1, Rk(Pic) = 3, Orbits [ 1, 1, 1, 8, 8, 8 ] 250 #U = 64 [ 2, 2, 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 2, 8, 16 ] 251 #U = 64 [ 2, 2, 2, 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 2, 4, 4, 16 ] 252 #U = 64 [ 2, 4 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 2, 8, 16 ] 253 #U = 64 [ 2, 4 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 2, 8, 16 ] 254 #U = 64 [ 2, 2, 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 2, 8, 16 ] 255 #U = 64 [ 2, 4 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 2, 8, 16 ]

appendix 245

256 #U = 72 [ 2, 2 ], #H^1 =4[2,2],Rk(Pic) = 1, Orbits [ 6, 6, 6, 9 ] 257 #U = 72 [ 2, 2 ], #H^1 = 2, Rk(Pic) = 1, Orbits [ 6, 9, 12 ] 258 #U = 72 [ 2, 2, 2 ], #H^1 = 2, Rk(Pic) = 1, Orbits [ 6, 9, 12 ] 259 #U = 72 [ 2, 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 6, 6, 6, 9 ] 260 #U = 72 [ 3, 3 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 3, 24 ] 261 #U = 72 [ 2, 2 ], #H^1 = 2, Rk(Pic) = 1, Orbits [ 6, 9, 12 ] 262 #U = 72 [ 2, 4 ], #H^1 = 2, Rk(Pic) = 1, Orbits [ 6, 9, 12 ] 263 #U = 72 [ 2, 2, 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 3, 3, 6, 6, 9 ] 264 #U = 72 [ 2, 2, 2 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 3, 6, 18 ] 265 #U = 80 [ 5 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 10, 16 ] 266 #U = 81 [ 3, 3 ], transitive 267 #U = 96 [ 3 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 3, 24 ] 268 #U = 96 [ 3 ], #H^1 = 1, Rk(Pic) = 3, Orbits [ 1, 1, 1, 8, 8, 8 ] 269 #U = 96 [ 2, 2, 3 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 3, 24 ] 270 #U = 96 [ 2, 2, 2 ], #H^1 = 2, Rk(Pic) = 2, Orbits [ 1, 4, 6, 8, 8 ] 271 #U = 96 [ 2, 2, 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 2, 2, 6, 16 ] 272 #U = 96 [ 4 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 3, 24 ] 273 #U = 96 [ 2, 2 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 3, 24 ] 274 #U = 96 [ 2, 2, 3 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 4, 6, 16 ] 275 #U = 96 [ 2, 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 4, 6, 16 ] 276 #U = 96 [ 2, 4 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 4, 6, 16 ] 277 #U = 96 [ 2, 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 4, 6, 16 ] 278 #U = 96 [ 2, 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 4, 6, 16 ] 279 #U = 96 [ 2, 2, 2 ], #H^1 = 2, Rk(Pic) = 1, Orbits [ 3, 12, 12 ] 280 #U = 108 [ 2, 2 ], #H^1 = 3, Rk(Pic) = 1, Orbits [ 9, 9, 9 ] 281 #U = 108 [ 2, 2, 3 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 9, 18 ] 282 #U = 108 [ 2, 2 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 9, 18 ] 283 #U = 108 [ 2, 2, 3 ], #H^1 = 3, Rk(Pic) = 1, Orbits [ 9, 9, 9 ] 284 #U = 108 [ 2, 2 ], #H^1 = 3, Rk(Pic) = 1, Orbits [ 9, 9, 9 ] 285 #U = 108 [ 4 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 9, 18 ] 286 #U = 108 [ 4 ], transitive 287 #U = 108 [ 3, 4 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 9, 18 ] 288 #U = 108 [ 2, 2 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 9, 18 ] 289 #U = 108 [ 2, 2 ], transitive 290 #U = 108 [ 2, 2, 3 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 9, 18 ] 291 #U = 120 [ 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 6, 6, 15 ] 292 #U = 120 [ 2 ], #H^1 = 2, Rk(Pic) = 1, Orbits [ 12, 15 ] 293 #U = 120 [ 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 2, 5, 10, 10 ] 294 #U = 120 [ 2 ], #H^1 = 1, Rk(Pic) = 3, Orbits [ 1, 1, 5, 5, 5, 10 ] 295 #U = 120 [ 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 2, 5, 10, 10 ] 296 #U = 120 [ 2 ], #H^1 = 2, Rk(Pic) = 1, Orbits [ 12, 15 ] 297 #U = 128 [ 2, 2, 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 2, 8, 16 ] 298 #U = 144 [ 2, 2, 2 ], #H^1 = 2, Rk(Pic) = 1, Orbits [ 6, 9, 12 ] 299 #U = 144 [ 2 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 3, 24 ] 300 #U = 160 [ 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 10, 16 ] 301 #U = 162 [ 2, 3 ], transitive 302 #U = 162 [ 2, 3 ], transitive 303 #U = 162 [ 2 ], transitive 304 #U = 192 [ 2 ], #H^1 = 1, Rk(Pic) = 3, Orbits [ 1, 1, 1, 8, 8, 8 ] 305 #U = 192 [ 2 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 3, 24 ] 306 #U = 192 [ 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 2, 8, 16 ] 307 #U = 192 [ 2 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 3, 24 ] 308 #U = 192 [ 2, 3 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 2, 8, 16 ] 309 #U = 192 [ 2, 3 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 3, 24 ] 310 #U = 192 [ 2, 2 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 3, 24 ] 311 #U = 192 [ 2, 2, 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 4, 6, 16 ] 312 #U = 216 [ 2, 2, 2 ], #H^1 = 3, Rk(Pic) = 1, Orbits [ 9, 9, 9 ] 313 #U = 216 [ 2, 2 ], transitive 314 #U = 216 [ 2, 4 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 9, 18 ] 315 #U = 216 [ 2, 2 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 9, 18 ] 316 #U = 216 [ 2, 2 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 9, 18 ] 317 #U = 216 [ 2, 2 ], transitive 318 #U = 216 [ 2, 2, 2 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 9, 18 ] 319 #U = 216 [ 8 ], transitive 320 #U = 216 [ 2, 2 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 9, 18 ] 321 #U = 216 [ 2, 2, 3 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 9, 18 ] 322 #U = 240 [ 2, 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 2, 5, 10, 10 ] 323 #U = 240 [ 2, 2 ], #H^1 = 2, Rk(Pic) = 1, Orbits [ 12, 15 ] 324 #U = 288 [ 3, 3 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 3, 24 ] 325 #U = 320 [ 4 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 10, 16 ] 326 #U = 324 [ 3 ], transitive 327 #U = 324 [ 2, 2 ], transitive 328 #U = 360 [ ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 6, 6, 15 ] 329 #U = 384 [ 2, 2 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 3, 24 ] 330 #U = 384 [ 2, 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 2, 8, 16 ] 331 #U = 432 [ 2, 2, 2 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 9, 18 ] 332 #U = 432 [ 2, 2 ], transitive 333 #U = 576 [ 2 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 3, 24 ] 334 #U = 576 [ 2, 3 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 3, 24 ] 335 #U = 576 [ 2, 3 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 3, 24 ] 336 #U = 648 [ 2 ], transitive 337 #U = 648 [ 2 ], transitive 338 #U = 648 [ 2, 3 ], transitive

246 appendix

339 #U = 648 [ 3 ], transitive 340 #U = 720 [ 2 ], #H^1 = 2, Rk(Pic) = 1, Orbits [ 12, 15 ] 341 #U = 720 [ 2 ], #H^1 = 2, Rk(Pic) = 1, Orbits [ 12, 15 ] 342 #U = 720 [ 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 6, 6, 15 ] 343 #U = 960 [ ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 10, 16 ] 344 #U = 1152 [ 2, 2 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 3, 24 ] 345 #U = 1296 [ 2 ], transitive 346 #U = 1296 [ 2, 2 ], transitive 347 #U = 1440 [ 2, 2 ], #H^1 = 2, Rk(Pic) = 1, Orbits [ 12, 15 ] 348 #U = 1920 [ 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 10, 16 ] 349 #U = 25920 [ ], transitive 350 #U = 51840 [ 2 ], transitive

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Õ×Ð -ÐÞ ÒÕÒÓÑÙ ÖÙ,Ù¸ Î×¬ º ÆÚÙ ËËÊ¸ ËÖº ¬Þº¹ÅØº

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Index

abelian surface, 196 AMD, 182 abelian variety, 41, 166 Amir-Moez, A. R., 83 Abramovich, D., 39 Arakelov, S. Yu., 25 accumulating subvariety, 9, 36, 195, 196, Arakelov degree, 6, 22 199 Archimedean valuation, 17 accumulating surface, 196, 199 ARIBAS, 183 adelic intersection product, 33 arithmetic cohomology, 83 adelic metric, 27, 31, 45, 46 arithmetic degree, 30 adelic Picard group, 25, 27, 30, 33 arithmetic intersection theory, 21 adelic point, 4–8, 10, 56, 61, 119–121, 124, arithmetic Picard group, 21 137, 145, 206, 208, 231 arithmetic variety, 6, 21, 23–25 adelic topology, 121 Artin conductor, 215, 236 adelically metrized invertible sheaf, 27–31, Artin L-function, vii, 51, 207, 208, 216, 33 222, 231 adjunction formula, 55 assembly, 179 affine cone, 54 Auslander–Goldman, theorem of, 7, 98, algebraic Brauer–Manin obstruction, 5, 132 103, 105 algebraic surfaces, 98, 106 automorphisms of PN ,20 classification of, 167 Azumaya algebra, 83, 89–98, 104–106, 123, algorithm 124, 128 FFT point counting, 190, 192, 200 for computation of volume, 50 for numerical integration, 193 bad obvious, 189 prime, 192 of Tate, 159 reduction, 153 to compute approximate value for Batyrev, V. V., 33, 40, 66, 78, 79 Peyre’s constant, 192 Batyrev and Manin, conjecture of, 6, 38, to detect conics on quartic threefold, 198 40–43, 166, 196 to detect lines on cubic threefold, 196, Bernstein, D., 166, 199 200 to search for solutions of Diophantine Bezout’s theorem, 194 equation, 9, 169–182, 200 Bhowmik, G., 80 naive, 169 bielliptic surface, 166, 196 to solve system of equations over finite Bierce, A., 165 field, 196, 198 Billard, H., 41 to test for conic through two points, 197 biquadratic reciprocity low, 4 to test lines for irrationality, 195 Bogomolov, F., 42 to test lines for irrationality, 196 conjecture of, 42 to test whether conic is contained in Bombieri, E., 47 quartic threefold, 198 Bourbaki, N., 29 almost Fano variety, 66 Brauer, R., 84

261

262 index

Brauer group, 4–7, 83, 84, 94, 98, 99, congruence, 2, 9, 60, 161, 167–178, 180, 101–103, 105–117 181, 183, 235 cohomological, 7, 98–103, 105–117 conic, 197–199 of cubic surface, 116 conjecture Brauer–Manin obstruction, 5, 7–9, 36, 59, of Batyrev and Manin, 6, 38, 40–43, 166, 61, 117, 123–125, 129, 132, 140, 146, 196 160, 161, 187, 188, 206–208 of Bogomolov, 42 algebraic, 5, 132 of Colliot-Thélène, 125 is the only obstruction, 125 of Lang, 2, 6, 37–39, 166, 196 to rational points, 125 geometric, 38 to the Hasse principle, 8, 36, 61, 125, strong, 38 132, 140–142, 148 weak, 38, 40 to weak approximation, 10, 125, 140, of Manin, 3, 6, 38, 46, 47, 63, 65, 67, 165, 141, 145, 148 185–188, 197, 201, 202, 205 transcendental, 5, 132 of Manin and Peyre, 59, 61, 66, 67, 202 Brauer–Severi variety, 122 of Mordell, 38 Bremner, A., 4, 122 control function, 177, 180 Bright, M. J., 161, 166 convolution, 189, 190 Browning, T. D., 66, 67 Corn, P. K., 7, 117 Brüdern, J., 175 Cox ring, 64 bucket sort, 180 Cox, D. A., 64 buffer, 181, 182 cubic threefold, 9 cubic reciprocity low, 4, 161 C, 179, 183 cubic surface, 4–8, 10, 47–51, 56, 61, 62, Cantor, M., 239 115–117, 122, 125, 132, 140, 143–148, Cassels, J. W. S., vii, 4, 122 159, 160, 165, 193, 194, 197, 237, 239 Cayley’s cubic surface, 66 Cayley’s, 66 Čech cocycle, 96 diagonal, 10, 122, 146–148, 159, 205–216, Čech cohomology, 96, 97 237 center, 84, 89, 90 general, 125 central simple algebra, 84, 85, 89–91, 101, minimal, 234 103 cubic threefold, 9, 10, 39, 62, 185–203 index of, 84 diagonal, 9, 10, 36, 37, 47 that splits, 85, 86 cusp, 194 Chambert-Loir, A., 78 Chebotarev density theorem, 214, 236 Chebyshev’s inequalities, 235 de Jong, A. J., 106 Chinese remainder theorem, 177, 180, 181 de la Bretèche, R., 66, 79 circle method, 3, 5, 60–62, 187 decoupled equation, 9, 171, 199 class number formula, 223 Dedekind zeta function, 191, 223 classification of algebraic surfaces, 167 Del Pezzo surface, 50, 65, 66, 161 clustering, 171 weak, 66, 67, 79 cocycle, 86, 87 Deligne, P., 58, 168, 189 cohomologous to another, 86 Derenthal, U., 50, 64, 66, 67, 80 cocycle relations, 127 descent local in étale topology, 93 faithful flat, 93 cohomological Brauer group, 7, 98–103, Galois, 64, 87 105–117 descent method, 63–77 Colliot-Thélène, J.-L. , 5, 9, 61, 65, 122, Diophantine equation, 1, 2, 4, 6, 9, 15, 165, 125, 146–148, 206, 220, 221, 234 169–182, 199, 200 conjecture of, 125 decoupled, 9, 171, 199 Colliot-Thélène, Kanevsky, and Sansuc, Diophantus, 1, 239 Theorem of, 147, 148 Dirichlet character, 222 collision, 171 distance, 28, 29 conductor-discriminant formula, 215 Duke, W., 236

index 263

Eckardt point, 194 genus one curve, 137, 146, 153, 155, 156, Edidin, D., 107 166, 198, 199 effective cone, 47–49 Gilbert, W. S., 185 Eisenstein polynomial, 134 Gillet, H., 25, 33 elementary symmetric functions, 19, 212 global class field theory, 4, 107, 111 elliptic cone, 194 global evaluation map, see Manin map elliptic curve, 154 gluing data, 93, 96, 127 elliptic surface, 166 Godeaux surface, 39 Elsenhans, A.-S., 165, 185, 205 good Emerson, R. W., 205 reduction, 168, 211, 213, 227 Enriques surface, 110 Gröbner base, 144, 145 equidistribution, 5, 158, 188, 205 Grothendieck, A., 6, 83, 98, 106 Essouabri, D., 80 Guy, M. J. T., 4, 122 étale cohomology, 45, 57, 83, 102 étale morphism, 51, 53 Hankel, H., 1 étale neighbourhood, 96, 108, 111 Harari, D., 132 étale topology, 93, 96, 99, 101 hardware, 176 Euler product, 190, 192, 220, 221 Hardy, G. H., 3 Euler sequence, 55 Hartshorne, R., vii exponential sequence, 45, 109 hash function, 170, 177, 180 hash table, 171, 180, 181, 200 factor α, 4, 6, 47–50, 56, 60, 61, 187, 188, 207, 210, 231 hashing, 9, 170, 180, 222 factor β, 6, 50, 187, 207, 210, 231 uniform, 170 faithful flat descent, 93 Hasse principle, 3–5, 8, 122, 125, 132, 205, Faltings, G., 15, 25, 38 221 Fano variety, 2, 4, 6, 9, 37, 38, 40, 41, 43, Brauer–Manin obstruction to, 8, 36, 61, 44, 46, 47, 58, 59, 61, 165, 185, 186, 125, 132, 140–142, 148 188, 205 counterexample to, 8, 122, 141, 142 Fano variety Hasse’s bound, 137, 156 almost, 66 Hasse, H., 3, 205 Faraday, M., 119 Hassett, B., 107 Fermat cubic, 37 Hawking, S., 15 FFT convolution, 189 Heath-Brown’s congruences, 161 FFT point counting, 192, 200 Heath-Brown, D. R., 5, 61, 62, 66–68, 79, Forster, O., 183 161, 165, 233 Fouvry, È, 79 Hecke L-function, 216 fpqc-topology, 93 Hecke theta function, 217, 218 Fröhlich, A., vii height Franke, J., 3, 47, 77, 78 absolute, 18, 20, 30, 45 Frei, F., 80 adelic, 29 Frobenius eigenvalues, 58, 212 anticanonical, 3, 40, 45, 59, 205 Fubini-Study metric, 22, 24 canonical, 3, 40, 45, 59, 205 fundamental finiteness, 6, 10, 15, 16, 25, defined by adelic metric, 45 209, 222, 233 defined by an invertible sheaf, 6, 20 l2,24 G-set, 114 logarithmic, 20, 23 Gabber, O., 106 naive, 3, 6, 15, 16, 18, 23, 186–188, 205, Galois cohomology, 83 233 Galois descent, 64, 87 of smallest point, 10 GAP, 7, 48, 49, 116, 144, 210, 239, 241 with respect to hermitian line bundle, 6, Gauß–Legendre formula, 193 23 Gauß sum, 213 Hensel’s lemma, 2, 52, 94–96, 102, 124, 133, Generalized Riemann Hypothesis, 233, 236, 137, 153, 157, 173, 235 237 Hensel, K., 2

264 index hermitian line bundle, 6, 21–24, 46 Lefschetz theorem on (1, 1)-classes, 109 continuous, 22 Lefschetz trace formula, 57, 211, 213 smooth, 22 Legendre’s theorem, 3, 205 hermitian metric, 21, 24–26, 32, 187 Legendre, A.-M., 3, 205 bounded, 27 lemma continuous, 21, 27, 46, 188 of Hensel, 2, 52, 94–96, 102, 124, 133, smooth, 27 137, 153, 157, 173, 235 heuristic, 35 of Wedderburn and Brauer, 84 Hirzebruch, F., 189 of Yoneda, 126 Hochschild–Serre spectral sequence, 5, 83, of Zorn, 126 111, 130 Leray measure, 53, 55, 60, 62, 187, 188, hypersurface measure, 53, 227, 228 209, 227, 229 Lichtenbaum, S., 113 Ieronymou, E., 132 theorem of, 113 index Lichtenbaum duality, 113, 160 of central simple algebra, 84 Lichtin, B., 80 induced Lind, C.-E., 4, 5, 122 character, 213, 214 line, 1, 2, 8, 36, 37, 39, 43, 48–50, 133, representation, 213, 214 136–140, 143, 144, 146, 148, 153, 186, inﬂation, 130, 131, 145, 151 194–200 intersection of two sets, 169 non-obvious, 10, 39, 196, 200 invariant map, 107 obvious, 9, 39, 194, 196 irrationality test for lines, 195, 196 sporadic, 10, 39, 196, 200 –s, 27 on cubic surface, see 27 lines on Jacobi sum, 135, 190, 213 cubic surface Janda, F., 80 linear probing, 171 linear subspace, 186 K3 surface, 9, 41–43, 110, 115, 167, 196 Kanevsky, D., 9, 61, 122, 146–148, 206, Linnik’s theorem, 233 220, 234 Littlewood, J. E., 3 Kelvin, W. Thomson 1st Baron, 35 local evaluation map, 123, 147, 152 Khayyám, O., 83 local measure, 52 Kodaira classiﬁcation, 37, 196 log-factor, 3, 47 Kodaira dimension, 196 Loughran, D., 66 Kresch, A., 107 Lovasz, L., 50 Kummer pairing, 214 Kummer sequence, 45, 100, 108, 109, 112 Maclagan-Wedderburn, J. H., 84 Kummer surface, 41 magma, 161 Manin, Yu. I., 3, 4, 8, 33, 40, 47, 67, 77, 78, L-function, vii, 51, 166, 207, 208, 216–219, 113, 117, 123, 132, 219, 220 222, 231, 236 Manin map, 7, 123, 129, 152 L1 cache, 178, 179, 182 Manin’s conjecture, 3, 6, 38, 46, 47, 63, 65, L2 cache, 179, 182 67, 165, 185–188, 197, 201, 202, 205 l3-unit sphere, 228 Manin’s formula, 113 Laface, A., 64 Manin–Peyre conjecture, 59, 61, 66, 67, 202 Lang’s conjecture, 2, 6, 37–39, 166, 196 maple, 136 geometric, 38 Maxwell, J. C., 119 strong, 38 McKinnon, D., 41 weak, 38, 40 Mellin transform, 217 Lang, S., 2, 6, 37–40, 166, 196 memory architecture, 178 Langian exceptional set, 38, 39 Mergesort, 180 Larsen, K., 15 metric, 26 lattice basis reduction, 142 bounded, 27 Le Boudec, P., 66 continuous, 27 Lebesgue measure, 47 induced by a model, 26 Lefschetz hyperplane theorem, 110 Milne, J., vii, 63, 83

index 265 minimum metric, 6, 22, 23, 188 polymake,50 Minkowski, H., 3 Poonen, B., 125 model, 6, 25–28, 43, 46, 52, 57, 121, 122, post-processing, 178 124, 127, 128, 146, 152, 153, 156, 158, prefetch, 179 159, 187, 194, 208 presorting, partial, 180 modular operation, 177 Preu, T., 132 Monte Carlo method, 50 principal divisor Mordell’s conjecture, 38 norm of, 154 Mordell’s examples of cubic surfaces, 4, product formula, 16 122, 132, 146 projective plane, 198 Mordell, L. J., 4, 8, 38, 122, 132 Puiseux expansion, 193 Moroz, B. Z., 79 pullback Murre and Oort, theorem of, 108 of Azumaya algebra, 91 pure cubic ﬁeld Néron–Severi group, 108, 112 discriminant of, 215, 237 Néron–Severi torus, 63, 64 Noether–Lefschetz Theorem, 5, 186, 193 quadratic form, 174–176, 178 non-Archimedean valuation, 17 quadratic reciprocity low, 4 non-Azumaya locus, 90, 106, 128 quadric, 198 non-obvious line, 10, 39, 196, 200 quartic surface normalized valuation, 16 diagonal, 42 Northcott’s theorem, 18 quartic threefold, 9, 10, 62, 185–207 numerical integration, 193 diagonal, 9, 10, 37, 47 Quicksort, 180 obvious line, 9, 39, 194, 196 open addressing, 171 Radix sort, 180 Opteron processor, 182 rational points order, 103–105 Brauer–Manin obstruction to, 125 maximal, 103–106 rational surface, 110, 112, 113, 115, 196 p-adic measure, 51, 52 rational variety, 166 p-adic unsolvability, 36 reading, 175–177, 181, 182, 200 p-adic valuation, 223 reciprocity low p-adic numbers, 2 biquadratic, 4 page prime, 169, 170, 172, 177, 180 cubic, 4, 161 paging, 172 quadratic, 4 parametrization, 2 reduced trace, 103 Pari, 192 restriction, 151 partial presorting, 181 Riemann zeta function, 222 Pentium 4 processor, 183 ruled surface, 194, 196, 199 periodicity isomorphism, for cohomology of ruled variety, 166 cyclic group, 154 Perron’s formula, 73 Salberger, P., 65 Peyre, E., 6, 8, 9, 33, 40, 47, 59–63, 66, 78, Sansuc, J.-J., 9, 61, 65, 122, 146–148, 206, 165, 186, 187, 205, 208 220, 234 Peyre’s constant, 6, 9, 10, 47, 59–61, 63, Schappacher, N., 1, 239 165, 185, 187, 189, 200, 205, 206, 219, Schmidt, W. M., 63 231, 233 Schröer, S., 106 Peyre’s Tamagawa type number, see search bound, 9, 182, 200, 209 Peyre’s constant searching for rational points, see searching Picard functor, 108 for solutions of Diophantine equation Picard group, 112 searching for solutions of Diophantine Picard rank equation, 9, 169–182, 200 of cubic surface, 117 naive, 169 Picard scheme, 108 selection of bits, 176 Poincaré residue map, 55 Selmer, E. S., 37, 122

266 index

Serre, J.-P. suspicious theorem of, 86 pair, 197, 198 sheaf of Azumaya algebras, 6, 89 point, 195, 196 Siegel’s estimate, 223 Swinnerton-Dyer’s list, 151 Siegel, C. L., 63, 205, 223 Swinnerton-Dyer, Sir Peter, 4, 7–9, 68, 79, similarity 117, 122, 142, 151, 167, 172, 202 of Azumaya algebras, 93, 94, 98 of central simple algebras, 84 singular cohomology, 83 Tamagawa measure, 56, 188, 208 64-bit processor, 176 Tamagawa number, see Peyre’s constant Skolem–Noether theorem, 85, 95, 96 Tate, J., 159 Slater, J. B., 68 Tate cohomology, 115 Smart, N. P., 166 Tate’s algorithm, 159 solutions of Diophantine equation tensor field, 53 algorithm to search for, 9, 169–182, 200 test for conic through two points, 197 naive, 169 theorem sorting, 180, 222 Hilbert 90, 88, 99, 111, 131 Soulé, C., 25, 33 of Auslander and Goldman, 7, 98, 103, splitting field, 85, 102 105 sporadic line, 10, 39, 196, 200 of Bezout, 194 Stark, H. M., 223 of Colliot-Thélène, Kanevsky, and statistical parameters, 201, 202 Sansuc, 147, 148 Steiner surface, 199 of Legendre, 3, 205 Strauch, M., 78 of Lichtenbaum, 113 Sullivan, A., 185 of Linnik, 233 surface, 47, 106, 115, 193 of Murre and Oort, 108 abelian, 196 of Noether and Lefschetz, 186, 193 algebraic, 98, 106 of Northcott, 18 bielliptic, 166, 196 of Serre, 86 cubic, 4–8, 10, 47–51, 56, 61, 62, of Skolem and Noether, 85, 95, 96 115–117, 122, 125, 132, 140, 143–148, of Tietze, 25, 32 159, 160, 165, 193, 194, 197, 237, 239 of Tsen, 110 Cayley’s, 66 of Zak, 195 diagonal, 10, 122, 146–148, 159, weak Lefschetz, 189 205–216, 237 Thompson, S., 35 general, 125 threefold, 36, 206 minimal, 234 cubic, 9, 10, 39, 62, 185–203 Del Pezzo, 50, 65, 66, 161 diagonal, 9, 10, 36, 37, 47 weak, 66, 67, 79 quartic, 10, 62, 185–207 elliptic, 166 diagonal, 9, 10, 37, 47 Enriques, 110 Tietze’s theorem, 25, 32 Godeaux, 39 toric variety, 78, 79 3 in P ,9 torsor, universal, 63–71 K3, 9, 41–43, 110, 115, 167, 168, 196 transcendental Brauer–Manin obstruction, Kummer, 41 5, 132 non-minimal, 43 Tschinkel,Y.,3,6,8,47,61,62,66,67, non-separated, 107 77–79, 165 of general type, 39, 195, 196 Tsen’s theorem, 110 of Kodaira dimension one, 196 27 lines on cubic surface, 7, 48–51, of Kodaira dimension zero, 41 115–117, 125, 144–146, 149, 160, rational, 110, 112, 113, 115, 196 209–213, 234, 239 ruled, 194, 196, 199 Tychonov topology, 121 Steiner, 199 surfaces classification of, 167 universal torsor, 63–71

index 267

Vallino, M., 221 valuation Archimedean, 17 lying above another, 17 non-Archimedean, 17 variety, 2 Fano,2,4,6,9,37,38,40,41,43,44,46, 47, 58, 59, 61, 165, 185, 186, 188, 205 almost, 66 of general type, 2, 37–40, 166, 195, 196 of intermediate type, 2, 37, 38, 41, 87, 88 toric, 78, 79 Vaughan, R. C., 63 Velaso, M., 64 Vistoli, A., 107 Vo jta, P., 42 Voloch, J. F., 39 weak approximation, 5, 8, 10, 61, 122, 125, 145 Brauer–Manin obstruction to, 10, 125, 140, 141, 145, 148 counterexample to, 5 weak Del Pezzo surface, 66, 67, 79 weak Lefschetz theorem, 189 Weil, A., 6, 168 Weil conjectures, 58, 168, 189 for curves, 168, 176 Weyl, H., 62 writing, 176–178, 180, 182, 200

Yoneda’s lemma, 126

Zak’s theorem, 195 Zhang, S., 33 Zorn’s Lemma, 126

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For a complete list of titles in this series, visit the AMS Bookstore at www.ams.org/bookstore/survseries/.

The central theme of this book is the study of rational points on algebraic varieties of Fano and intermediate type—both in terms of when such points exist and, if they do, their quantitative density. The book consists of three parts. In the first part, the author discusses the concept of a height and formulates Manin’s conjecture on the asymptotics of rational points on Fano varieties. The second part introduces the various versions of the Brauer group. The author explains why a Brauer class may serve as an obstruction to weak approximation or even to the Hasse principle. This part includes two sections devoted to explicit computations of the Brauer–Manin obstruction for particular types of cubic surfaces. The final part describes numerical experiments related to the Manin conjecture that were carried out by the author together with Andreas-Stephan Elsenhans. The book presents the state of the art in computational arithmetic geometry for higher- dimensional algebraic varieties and will be a valuable reference for researchers and graduate students interested in that area.

For additional information and updates on this book, visit AMS on the Web www.ams.org/bookpages/surv-198 www.ams.org

SURV/198