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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 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 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Permissions to reuse portions of AMS publication content are handled by Copyright Clearance Center’s RightsLink service. For more information, please visit: http://www.ams.org/rightslink. Send requests for translation rights and licensed reprints to [email protected]. Excluded from these provisions is material for which the author holds copyright. In such cases, requests for permission to reuse or reprint material should be addressed directly to the author(s). Copyright ownership is indicated on the copyright page, or on the lower right-hand corner of the first page of each article within proceedings volumes. c 2014 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321 191817161514 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 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 mathemat- ics, 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 al- gorithm. 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.
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