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Home , Algebraic geometry, Diophantine geometry, Height function

Fundamentals of Diophantine

Fundamentals of

Springer-Verlag Serge Lang Department of Yale University New Haven, CT 06520 U.S.A.

AMS Subject Classifications: lOB99, 14GXX

Library of Congress Cataloging in Publication Data Lang, Serge, 1927- Fundamentals of diophantine geometry. I. Diophantine analysis. 2. Geometry, Algebraic. I. Title. QA242.L235 1983 512'.74 83-361

An earlier version of this book, Diophantine Geometry, was published by Wiley• Interscience.

ISBN 978-1-4419-2818-4 ISBN 978-1-4757-1810-2 (eBook) DOI 10.1007/978-1-4757-1810-2

Published in 1983 by Springer Science+Business Media New York Origina11y published by Springer-Verlag New York Inc in 1983. Softcover reprint of the hardcover 1st edition 1983

Typeset by Composition House Ltd., Salisbury, England.

9 8 7 6 5 4 3 2 1 Foreword

Diophantine problems represent some of the strongest aesthetic attractions to . They consist in giving criteria for the existence of solutions of algebraic in rings and fields, and eventually for the of such solutions. The fundamental of interest is the ring of ordinary Z, and the fundamental of interest is the field Q of rational . One discovers rapidly that to have all the technical freedom needed in handling general problems, one must consider rings and fields of finite type over the integers and rationals. Furthermore, one is led to consider also finite fields, p-adic fields (including the real and complex numbers) as representing a localization of the problems under consideration. We shall deal with global problems, all of which will be of a qualitative nature. On the one hand we have defined over say the rational numbers. Ifthe is affine one may ask for its points in Z, and thanks to Siegel, one can classify all curves which have infinitely many integral points. This problem is treated in Chapter VII. One may ask also for those which have infinitely many rational points, and for this, there is only Mordell's conjecture that if the is :;;; 2, then there is only a finite number of rational points. On the other hand, we have abelian varieties. If A is an defined over Q, then its of rational points is finitely generated (Mordell• Weil theorem). The proofs do not give a constructive method to get hold of the generators. Abelian varieties include of course curves of genus 1, which have been the fundamental testing ground for conjectures, theorems, and proofs in the theory. As a curve of genus:;;; 1 is characterized by the fact that it is embeddable in an abelian variety, one is led to study subvarieties of abelian varieties, and especially Mordell's conjecture, in this light. A curve of genus :;;; 2 is vi Foreword characterized by the fact that it is unequal to any non-zero translation of itself in its Jacobian. This is a very difficult hypothesis to use. It is also interesting to look at subvarieties of other group varieties. F or us, the natural ones to consider are the toruses, i.e. products of multiplicative groups. It will be shown that ifG ls such a group, Va curvecontained in G, ra finitely generated subgroup of G, and V n r is infinite then Vis the translation of a subtorus (in 0). For various generalizations, and comments on the above results and problems, see Chapters VI and VII and the Historical Notes at the end ofthese chapters. The prototypes of the diophantine results and methods given in this book were developed by MordelI, Weil, and Siegel between 1920 and 1930. Considering recent developments in the techniques of algebraic geometry, and Roth's theorem, it seemed quite worth while to give a systematic ex• position ofthe theory as it stands, i.e. essentially a qualitative theory. One of the directions of research at present lies in making theorems quantitative, that is, getting estimates for the number of rational or integral points, or the generators of the Mordell-Weil group. It is a striking feature of the proofs that they are highly non-constructive. There do exist still some unsolved qualitative problems. Aside from Mordell's conjecture, for instance, I would conjecture that Siegel's result on integral points remains true for an affine open subset of an abelian variety, defined over Z (or a ring of finite type over Z). In general, the extension of Roth's theorem and the theory of integral points as given in Chapter VII to varieties of > 1 is stilliacking.

New York, 1961 S.LANG

[The above is reproduced verbatimfrom Diophantine Geometry. Chapters VI and VII are now included in Chapters 7 and 8 respectively.] Preface

Diophantine Geometry has been out of print for a while. Advances in algebraic geometry, especially in some ofthe problems ofdiophantine geometry, have motivated me to bring out a new book, and I thank Springer-Verlag for publishing it. Three years after Diophantine Geometry appeared, Neron published his fundamental paper giving his canonical heights and explaining how they could be decomposed as sums of local functions. Tate also gave his simple global existence proof for the canonical height. These constitute the main new topics which I treat here. I have also included the Chevalley-Weil theorem; Mumford's theorem concerning heights on curves; Liardet's theorem on the interseetion of curves with division points; Schanuel's counting of points in projective ; and theorems of Silverman and Tate on heights in algebraic families. I still emphasize the Mordell-Weil theorem, Thue-Siegel-Roth theorem, Siegel's theorem on integral points and related results, and the Hilbert irreducibility theorem. These basically depend on heights rather than Neron functions, and so the material on Neron functions (as distinguished from the canonical heights) has been put at the end. Twenty years ago, the modern period of algebraic geometry was in its infancy. Today the situation is different. Several major advances have not been included although they would be equally worthy. They deserve books to themselves, and I list some of them. Neron's paper proving the existence of a minimal model for abelian varieties was already a book, and only a full new book could do justice to a new exposition of his result and some of its applications; these include Spencer Bloch's interpretation of the Birch• Swinnerton-Dyer conjecture together with its relation to the Neron pairing. I have not included Parsin's and Arakelov's rigidity theorems. I have not viii Preface included Manin's proof of the MordeIl conjecture over fields; nor Grauert's sub se quent other proof, depending on differential geometric con• siderations on the curve; nor Parsin's proof of Manin's theorem based on entirely different considerations. I have not included Raynaud's proofthat the of a curve with the torsion in an abelian variety is finite, nor Bogomolov's resuIts in this same direction. The methods of proof depend on group schemes and p-adic representations. I have not included Arakelov's on . Such results and others would fill several , as a continuation of the present book, which merely lays down the basic concepts of diophantine geometry. And the resuIts mentioned above are some of those with a unity of style relative to the topics chosen here. This leaves out transcendental methods, the in many variables (quasi-algebraic closure, methods), etc. When Diophantine Geometry first came out in 1961, some people regarded methods of algebraic geometry in as something rather far out, and some mathematicians objected to the general style, as when MordeIl wrote (BulI. AMS (1964), p. 497): " ... When proof of an extension makes it exceedingly difficult to understand the simpler cases, it might sometimes be better ifthe generalizations were left in the Journals." (Mordell's review and my review of his book are reproduced as an appendix.) Of course, a question here is: "exceedingly difficuIt" to whom? When? MordeIl wrote that he feIt like Rip van Winkle. But great progress in mathematics is often accompanied by such feelings. An algebraic geometer who went to sleep in 1961 and woke up in 1981 also might feellike Rip van Winkle. At the time of the first edition, the foundations of algebraic geo• metry were about to shift from the van der Waerden-Chevalley-Weil• Zariski methods to the Grothendieck methods. Today we have a more extensive perspective. On the whole, neither will bury the other. They serve different purposes, and even complementary purposes. Just as it is essential to reduce mod p2 (as in Raynaud's proof), and thereby rely on the full power of schemes and commutative , it is also essential to compute estimates on the coefficients and degrees of algebraic operations, as in the theory of transcendental numbers and algebraic independence several years ago, and in recent work of Wustholz and others. For these applications, one needs the older coordinates (I was about to write of van der Waerden• Chow-Chevalley-Weil-Zariski) of Lasker, Gordan and the nineteenth century algebraic geometers. Anyone who rejects any part of these contribu• tions does so at their own peril. Furthermore, the fact that estimates of coefficients in algebraic operations occurred during recent times first in the direction of transcendental numbers does not mean that this direction is the only application for them. It is a legitimate, and to many people an interesting ofview, to ask that the theorems of algebraic geometry from the Hilbert Nullstellensatz to the most advanced results should carry with them estimates on the coefficients occurring in these theorems. AIthough some of the estimates are routine, serious and interesting problems arise in this context. Preface IX

As in Diophantine Geometry the eontent of the present is still essentially qualitative. My eonjeeture that there is only a finite number of integral points on an affine open sub set of an abelian variety is still unproved. Roth's theorem is still not effeetive, and the main advanee has been due to Sehmidt's generalizations to higher . My introduetion to Elliptic Curves: Diophantine Analysis makes it unneeessary to diseuss he re onee more tlie relative roles of general theorems and eonerete examples or special eases, whieh are eomplementary. The present book is addressed to those whose taste lies with abelian varieties.

New Haven, 1983 S.LANG

As this book goes to press, Fa/tings has announced his proof of the Mordell conjecture. Arbeitstagung, Bonn June 1983 Contents

Acknowledgment . . . . xv Some Standard Notation xvii

CHAPTER I Absolute Values 1. Definitions, dependence and independence 1 2. Completions. . . . . 5 3. Unramified extensions. 9 4. Finite extensions 12

CHAPTER 2 Proper Sets of Absolute Values. Divisors and Units ..... 18 1. Proper sets of absolute values. 18 2. Number fields . . . 19 3. Divisors on varieties . . . . 21 4. Divisors on schemes. . . . . 24 5. MK-divisors and divisor classes . 29 6. classes and units in number fields. 32 7. Relative units and divisor classes 41 8. The Chevalley-Weil theorem. 44

CHAPTER 3 Heights .. 50 1. Definitions. . 50 2. Gauss' lemma 54 3. Heights in function fields 62 XlI Contents

4. Heights on abelian groups 66 5. Counting points of bounded height 70

CHAPTER 4 Geometrie Properties of Heights . 76 1. Funetorial properties . . . 76 2. Heights and linear systems . 83 3. Ample linear systems . . . 87 4. Projeetions on eurves . . . 90 5. Heights assoeiated with divisor c1asses . 91

CHAPTER 5 Heights on Abelian Varieties 95 1. Some linear and quasi-. 95 2. Quadratieity of endomorphisms on divisor c1asses . 99 3. Quadratieity of the height . . 106 4. Heights and Poineare divisors . . . . . 110 5. Jaeobian varieties and eurves...... 113 6. Definiteness properties Over number fields 120 7. Non-degenerate heights and Euclidean spaees . 124 8. Mumford's theorem...... 134

CHAPTER 6 The Mordell-Weil Theorem. 138 1. Kummer theory ...... 139 2. The weak Mordell-Weil theorem 144 3. The infinite deseent . . . 145 4. Reduetion steps. . . . . 146 5. Points of bounded height 149 6. Theorem of the base 153

CHAPTER 7 The Thue-Siegel-Roth Theorem 158 1. Statement of the theorem. . . . 158 2. Reduetion to simultaneous approximations 163 3. Basie steps of the proof . 165 4. A eombinatoriallemma. 170 5. Proof of Proposition 3.1 171 6. Wronskians...... 173 7. Faetorization of a polynomial. 175 8. The index...... 178 9. Proof of Proposition 3.2 . . . 181 10. A geometrie formulation of Roth's theorem 183 Contents xiii

CHAPTER 8 Siegel's Theorem and Integral Points 188 1. Height of integral points. 189 2. Finiteness theorems. 192 3. The curve ax + by = 1 194 4. The Thue-Siegel curve. 196 5. Curves of genus 0 . 197 6. Torsion points on curves. 200 7. Division points on curves 205 8. Non-abelian Kummer theory . 212

CHAPTER 9 Hilbert's Irreducibility Theorem. 225 I. Irreducibility and integral points . 226 2. Irreducibility Over the rational numbers 229 3. Reduction steps ...... 233 4. Function fields...... 236 5. Abstract definition of Hilbert sets. . . 239 6. Applications to commutative group varieties 242

CHAPTER 10 Weil Functions and Neron Divisors 247 1. Bounded sets and functions . . . 247 2. Neron divisors and Weil functions 252 3. Positive divisors ...... 258 4. The associated . . 263

CHAPTER 11 Neron Functions on Abelian Varieties . 266 1. Existence of Neron functions...... 266 2. Translation properties of Neron functions 271 3. Neron functions on varieties ...... 276 4. Reciprocity laws ...... 283 5. Neron functions as intersection multiplicities 286 6. The Neron symbol and group extensions. . 290

CHAPTER 12 Algebraic Families of Neron Functions 296 1. Variation of Neron functions in an algebraic family 297 2. Silverman's height and specialization theorems 303 3. Neron heights as intersection multiplicities. . . . 307 4. Fibral divisors ...... 314 5. The height determined by a seetion : Tate's theorem 320 xiv Contents

CHAPTER 13 Neron Functions Over the Complex Numbers 324 I. The Neron function of an abelian variety. . . . 324 2. The scalar product of differentials of first kind . 327 3. The canonical 2-form and the Riemann theta function 332 4. The divisor of the Riemann theta function . . . . 334 5. Green, Neron, and theta functions ...... 339 6. The law of interchange of argument and parameter 341 7. Differentials ofthird kind and Green's function .. 344

Appendix ...... 347 Review of S. Lang's Diophantine Geometry, by L. J. MordelI. 349 Review ofL. J. Mordell's Diophantine Equations, by S. Lang. 355

Bibliography ...... 359

Index...... 367 Acknowledgment

I thank Michel Laurent and Michel Waldschmidt for useful comments. I am especially indebted to Joe Silverman for his thorough going over of the manuscript and a long list of valuable suggestions and corrections. I am also indebted to Silverman and Tate for their manuscripts which formed the basis for the next to last chapter. Some Standard Notation

The following notation is used in a standard way throughout.

Rings are assumed commutative and without divisors of 0, unless otherwise specified.

P group of all roots of unity. p(K) subgroup of roots of unity in a field K. R*, K* invertible elements in a ring R (resp. in a field K). Ka algebraic c10sure of a field K. K(P) field obtained by adjoining to K a set of affine coordinates for a point P (equal to K(xO/xi , ••• ,xn/xi) if (xo, ... ,xn) are projective coordinates for P). R(a) R/a for any ideal a. ZeN) Z/NZ. A(P) p-primary part of an abelian group A (that is, the subgroup of elements whose order is apower of p). Am subgroup of elements x in an abelian group A such that mx = O. [nJ multiplication by an n on an abelian group. V(K) set of K-valued points of a variety or V. D '" D' for divisors D, D', linear equivalence. D~D' for divisors D, D', algebraic equivalence. Div(V) group of divisors on a variety V. DiviV) subgroup of divisors algebraically equivalent to O. Divl(V) subgroup of divisors linearly equivalent to O. xviii Some Standard Notation

Pic(V) Div(V)/DivlV). Pico (V) Div i V)/Divl V). NS(V) Div(V)/Diva(V) (the Neron-Severi group of V). h '" h' equivalence for functions, Ih - h' I is bounded. h ';::; h' quasi-equivalence for functions: for each B > 0,

- Cl + (1 - B)h ~ h' ~ (1 + B)h + C2 • h 4, h' for functions, with h' positive, there exists a constant C > ° such that Ih I ~ Ch'. Same as h = O(h'). h ~4, h' both h, h' positive, h 4, h' and h' 4, h.

UsuaIly h, H denote heights with h = log H. These are indexed to specify qualifications:

height determined by a qJ into . height relative to a field K. height determined by a morphism derived from the linear system 2'(X), weIl defined up to 0(1). on an arbitrary variety, the height associated with a divisor dass c, determined only up to 0(1); on an abelian variety, the canonical height. canonical height if one needs to distinguish it from an equiva• lence dass of heights.

Standard references:

lAG = I ntroduction to Algebraic Geometry [L 2]. A V = Abelian Varieties [L 3].

Weil's Foundations is still quoted in canonical style, F2-Xy , Theorem Z, wh ich refers to the second edition. For schemes, see Hartshorne's Algebraic Geometry, and also Mumford's Abelian Varieties.