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

Home , Brauer group, Gaussian rational, Manin obstruction

Mathematical Surveys and Monographs Volume 198

Jörg Jahnel

American Mathematical Society Brauer Groups, Tamagawa Measures, and Rational Points on Algebraic Varieties

http://dx.doi.org/10.1090/surv/198

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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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

v 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 program- ming 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.

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[Is] Á×

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ËÖº ÅØº ´½ ./µ¸ .¾.ß.!=¸ English translation: Merkurjev, A. S.: Structure of the Brauer group of ﬁelds, Math.

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ÈØÖ ÙÖ º ÍÒÚº ÅØº Å"º ×ØÖÓÒÓÑº ½¿¹¾´½ ¿µ¸ /½ß/¿ English translation: Merkurjev, A. S.: Closed points of Brauer-Severi varieties, Vest-

nik St. Petersburg Univ. Math. 26-2(1993), 44–46

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,ÓÑÓÑÓÖÞÑ ÒÓÖÑÒÒÓ,Ó ÚÝÕØ¸ ÁÞÚº º ÆÙ ËËËÊ ËÖº ÅØº ´½ .¾µ¸ ½¼½½ß½¼!=¸ English translation: Merkurjev, A. S. and Suslin, A. A.: K-cohomology of Severi- Brauer varieties and the norm residue homomorphism, Math. USSR-Izv. 21(1983), 307–340 [deM/F] de Meyer, F. and Ford, T. J.: On the Brauer group of surfaces and subrings of k[x, y], in: van Oystaeyen, F. and Verschoren, A. (Eds.): Brauer groups in ring theory and algebraic geometry, Proceedings of a conference held at Antwerp 1981, Lecture Notes Math. 917, Springer, Berlin, Heidelberg, New York 1982, 211–221 [deM/I] de Meyer, F. and Ingraham, E.: Separable algebras over commutative rings, Lecture Notes Math. 181, Springer, Berlin, Heidelberg, New York 1971 [Mi] Milne, J. S.: Étale Cohomology, Princeton University Press, Princeton 1980 [Mol] Molien, T.: Über Systeme höherer complexer Zahlen, Math. Ann. 41(1893), 83–156 [Mord] Mordell, L. J.: On the conjecture for the rational points on a cubic surface, J. London Math. Soc. 40(1965), 149–158 [Mo-ta] Morita, Y.: Remarks on a conjecture of Batyrev and Manin, Tohoku Math. J. 49 (1997), 437–448 [Mo-ki] Moriwaki, A.: Intersection pairing for arithmetic cycles with degenerate Green cur- rents, E-print AG/9803054 256 bibliography

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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