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Home , Akshay Venkatesh, John Friedlander, Yuri Tschinkel, Zeev Rudnick

Equidistribution in , An Introduction NATO Science Series

A Series presenting the results of scientific meetings supported under the NATO Science Programme.

The Series is published by IOS Press, Amsterdam, and Springer in conjunction with the NATO Public Diplomacy Division

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The NATO Science Programme offers support for collaboration in civil science between scientists of countries of the Euro-Atlantic Partnership Council. The types of scientific meeting generally supported are “Advanced Study Institutes” and “Advanced Research Workshops”, and the NATO Science Series collects together the results of these meetings. The meetings are co-organized bij scientists from NATO countries and scientists from NATO’s Partner countries – countries of the CIS and Central and Eastern Europe.

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Series II: Mathematics, Physics and Chemistry – Vol. 237 Equidistribution in Number Theory, An Introduction

edited by University of Montreal,´ QC, Canada and Zeév Rudnick Tel-Aviv University,

Published in cooperation with NATO Public Diplomacy Division Proceedings of the NATO Advanced Study Institute on Equidistribution in Number Theory Montreal,´ Canada 11--22 July 2005

A C.I.P. Catalogue record for this book is available from the Library of Congress.

ISBN-10 1-4020-5403-3 (PB) ISBN-13 978-1-4020-5403-7 (PB) ISBN-10 1-4020-5402-5 (HB) ISBN-13 978-1-4020-5402-0 (HB) ISBN-10 1-4020-5404-1 (e-book) ISBN-13 978-1-4020-5404-4 (e-book)

Published by Springer, P.O. Box 17, 3300 AA Dordrecht, The Netherlands. www.springer.com

Printed on acid-free paper

All Rights Reserved © 2007 Springer No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. CONTENTS

Preface ix Contributors xi Biographical Sketches of the Lecturers xiii Andrew Granville and Zeev´ Rudnick/ Uniform Distribution 1 1 Uniform Distribution mod One 1 2 Fractional Parts of αn2 6 3 Uniform Distribution mod N 8 4 Normal Numbers 11 Andrew Granville and K. Soundararajan/ Sieving and the Erdos–Kac˝ Theorem 15 John B. Friedlander/ Uniform Distribution, Exponential Sums, and Cryptography 29 1 Randomness and Pseudorandomness 29 2 Uniform Distribution and Exponential Sums 30 3 Exponential Sums and Cryptography 31 4 Some Exponential Sum Bounds 34 5 General Modulus and Discrepancy of Diffie–Hellman Triples 35 6 Pseudorandom Number Generation 37 7 Large Periods and the Carmichael Function 41 8 Exponential Sums to General Modulus 45 9 Sums over Elliptic Curves 49 10 Proof Sketch of Theorem 4.1 51 K. Soundararajan/ The Distribution of Prime Numbers 59 1TheCramer´ Model and Gaps Between Consecutive Primes 59 2 The Distribution of Primes in Longer Intervals 67 3 Maier’s Method and an “Uncertainty Principle” 74 Andrew Granville and Zeev´ Rudnick/ Torsion Points on Curves 85 1 Introduction 85 2 A Proof Using Galois Theory 86 3 Polynomials Vanishing at Roots of Unity 88

v vi CONTENTS

Andrew Granville/ The distribution of roots of a polynomial 93 1 Introduction 93 2 Algebraic Numbers 96 3Ink Dimensions: the Bilu Equidistribution Theorem 98 4 Lower Bounds on Heights 100 5 Compact Sets with Minimal Energy 100 Emmanuel Ullmo/ Manin–Mumford, Andre–Oort,´ the Equidistrib- ution Point of View 103 1 Introduction 103 2 Informal Examples of Equi-Distribution 104 3 The Manin–Mumford and the Andre–Oort´ Conjecture 114 4 Equidistribution of Special Subvarieties 126 D. R. Heath-Brown / Analytic Methods for the Distribution of Rational Points on Algebraic Varieties 139 1 Introduction to the Hardy–Littlewood Circle Method 139 2 Major Arcs and Local Factors in the Hardy–Littlewood Cir- cle Method 146 3 The Minor Arcs in the Hardy–Littlewood Circle Method 154 4 Combining Analytic and Geometric Methods 160 Ulrich Derenthal and Yuri Tschinkel/ Universal Torsors over Del Pezzo Surfaces and Rational Points 169 1 Introduction 169 2 Geometric Background 176 3 Manin’s Conjecture 178 4 The Universal Torsor 179 5 Summations 183 6 Completion of the Proof 188 7 Equations of Universal Torsors 189 W. Duke/ An Introduction to the Linnik Problems 197 1 Introduction 197 2 The Linnik Problems 198 3 Holomorphic Modular Forms of Half-Integral Weight 200 4 Theta Series with Harmonic Polynomials 202 5 Linnik Problem for Squares and the Shimura Lift 203 6 Nontrivial Estimates for Fourier Coefficients 204 7Salie´ Sums 207 8 An Estimate of Iwaniec 209 9 Theorems of Gauss and Siegel 210 10 The Nonholomorphic Case (Duke, 1988) 211 CONTENTS vii

11 Transition to Subconvexity Bounds for L-Functions 212 12 An Application to Traces of Singular Moduli 212 Jens Marklof/ Distribution Modulo One and Ratner’s Theorem 217 1 Introduction 217 2 Randomness of Point Sequences mod 1 218 3 m√α mod One 224 4 mα mod One 232 5 Ratner’s Theorem 237 A. Venkatesh/ Spectral Theory of Automorphic Forms: A Very Brief Introduction 245 1 What Is a Homogeneous Space? 245 2 Spectral Theory: Compact Case 247 3 Dynamics 253 4 Spectral Theory: Noncompact Case 254 5 Hecke Operators 256 6 Gross Omissions: The Selberg Trace Formula 258 / Some Examples How to Use Measure Classification in Number Theory 261 1 Introduction 261 2 Dynamical Systems: Some Background 264 3 Equidistribution of n2α mod 1 267 4 Unipotent Flows and Ratner’s Theorems 269 5 Entropy of Dynamical Systems: Some More Background 281 6 Diagonalizable Actions and the Set of Exceptions to Little- wood’s Conjecture 286 7 Applications to Quantum Unique Ergodicity 295 S. De Bievre` / An Introduction to Quantum Equidistribution 305 1 Introduction 305 2 A Crash Course in Classical Mechanics 306 3 A Crash Course in Quantum Mechanics 319 4 Two Words on Semi-Classical Analysis 325 5 Quantum Mechanics on the Torus 326 Zeev´ Rudnick/ The Arithmetic Theory of Quantum Maps 331 1 Quantum Mechanics on the Torus 331 2 Quantizing Cat Maps 334 3 Quantum Ergodicity 337 4 Quantum Unique Ergodicity 339 5 Arithmetic QUE 340 Index 343 PREFACE

From July 11th to July 22nd, 2005, a NATO advanced study institute, as part of the series “Seminaire´ de mathematiques´ superieures”,´ was held at the Uni- versite´ de Montreal,´ on the subject Equidistribution in the theory of numbers. There were about one hundred participants from sixteen countries around the world. This volume presents details of the lecture series that were given at the school. Across the broad panorama of topics that constitute modern number the- ory one finds shifts of attention and focus as more is understood and better questions are formulated. Over the last decade or so we have noticed increas- ing interest being paid to distribution problems, whether of rational points, of zeros of zeta functions, of eigenvalues, etc. Although these problems have been motivated from very different perspectives, one finds that there is much in common, and presumably it is healthy to try to view such questions as part of a bigger subject. It is for this reason we decided to hold a school on “Equidistribution in number theory” to introduce junior researchers to these beautiful questions, and to determine whether different approaches can influence one another. There are far more good problems than we had time for in our schedule. We thus decided to focus on topics that are clearly inter-related or do not require a lot of background to understand. Since there were two major number theory research programs taking place during the academic year 2005–2006, on Analysis in number theory at the Centre de recherches mathematiques´ in Montreal,´ and during the spring semester 2006 on Rational and Integral Points on Higher-Dimensional Varieties at the Mathematical Sciences Re- search Institute in Berkeley, California, we decided to help prepare junior par- ticipants by inviting some lecturers who would go on to be senior participants at those programs. The lectures split into roughly ten topics (with lecturers): 1. The basics of uniform distribution (Granville, Rudnick). 2. Exponential Sums and cryptography (Friedlander, Granville). 3. Spectral Theory (Venkatesh).

ix x PREFACE

4. Hyperbolic geometry, and Ratner’s theorem (Marklof, Lindenstrauss, Venkatesh). 5. Quantum equidistribution (De Bievre,` Lindenstrauss, Rudnick, Venkatesh). 6. Distribution of integers and uncertainty principles (Soundararajan, Granville). 7. Distribution of rational points on varieties (Heath-Brown, Tschinkel). 8. Distribution of special points (Rudnick, Granville, Duke, Ullmo). 9. Spacing statistics (Marklof). 10. Invited lectures on relevant subjects (Harcos, Yafaev). The lecturers were requested to try to keep their lectures mostly self- contained (though they were allowed to require some light background read- ing before the meeting); in particular they were asked to avoid the use of very technical terms without giving independent motivation during their talks. In general the lecturers did so, and this was reflected by the high attendance throughout the meeting, despite the temptations of Montreal´ in mid-summer. We have asked the lecturers to bring over that attitude to the preparation of their contributions herein. We would like to thank the lecturers for their superb talks, and the gen- erosity with which they worked with the participants, as well as typed up their talks for these proceedings. We would also like to thank the Security through Science programme of NATO, the Centre de recherches mathematiques,´ the Institut des sciences mathematiques´ and the Universite´ de Montreal,´ for their generous and willing support of this school. The meeting would not have been possible without the organizational skills of Diane Belanger,´ for which we are very grateful. This book has bene- fitted from the typesetting skills of Louise Letendre and Andre´ Montpetit for which we thank them.

Andrew Granville Zeev´ Rudnick CONTRIBUTORS

Stephan De Bievre` Andrew Granville UFR de mathematiques´ Departement´ de mathematiques´ et Universite´ des Sciences et de statistique Technologies de Lille UniversitedeMontr´ eal´ 59655 Villeneuve d’Ascq C.P 6128, succ. Centre-ville France Montreal,´ QC H3C 3J7 [email protected] Canada lille1.fr [email protected] Ulrich Derenthal D. R. Heath-Brown Mathematisches Institut Oxford University Universitat¨ Gottingen¨ Mathematical Institute Bunsenstraße 3-5 24-29 St. Giles’ 37073 Gottingen¨ Oxford, OX1 3LB Germany UK [email protected] [email protected] goettingen.de Elon Lindenstrauss W. Duke Department of Mathematics Mathematics Department Fine Hall, Washington Road UCLA Box 951555 Princeton, NJ 08544 Los Angeles, CA 90095-1555 USA USA [email protected] Jens Marklof School of Mathematics John B. Friedlander University of Bristol Department of Computer and Bristol BS8 1TW Mathematical Sciences UK at Scarborough [email protected] 1265 Military Trail Toronto, ON M1C 1A4 Zeev Rudnick Canada School of Mathematical Sciences [email protected] Tel-Aviv University Schrieber Building, Room 316 Tel-Aviv 69978 Israel [email protected]

xi xii CONTRIBUTORS

K. Soundararajan Emmanuel Ullmo Arithmetique´ et Geom´ etrie´ Algebrique´ Mathematics, Bldg. 380 Universite´ -Sud 450 Serra Mall Batimentˆ 425 Stanford, CA 94305-2125 91405 Orsay Cedex [email protected] France [email protected] Yuri Tschinkel Akshay Venkatesh Mathematisches Institut Department of Mathematics Universitat¨ Gottingen¨ Courant Institute of Mathematical Science Bunsenstraße 3-5 251 Mercer Street 37073 Gottingen¨ New York, NY 10012-1185 Germany [email protected] [email protected] BIOGRAPHICAL SKETCHES OF THE LECTURERS

Stephan De Bi`evre is a mathematical physicist who received his education at the Universities of Antwerp, Leuven and Rochester. After a postdoctoral stay at the University of Toronto, and several years at the UniversiteParis´ 7, he became, in 1996, Professor of Mathematics at the Universite´ de Lille 1. He developed an interest in quantum chaos and hence in eigenfunction equidistribution a little more than a decade ago. William Duke received his PhD from the Courant Institute and was ’s third student. After being at UCSD for two years he moved to Rut- gers as an NSF Postdoc to work with . He was tenured at Rutgers and since 2000 he has been at UCLA. His research interests include the analytic theory of automorphic forms, L-functions, quadratic forms and elliptic curves. He has been especially interested in extensions and analogues of classical distribution problems from analytic number theory. John Friedlander received his doctorate at Penn State under the direction of S. Chowla and then did postdoctoral studies as Assistant to A. Selberg at the Institute for Advanced Study. He is currently University Professor of Math- ematics at the University of Toronto. His primary research interests are in analytic (especially multiplicative) and elementary (especially sieve-related) number theory. He is also interested in any part of mathematics which can say something about his favourite few problems, most of which are in some way related to quadratic polynomials. Andrew Granville, educated at Cambridge University in England and Queen’s University in Canada, is now the Canadian research chair in number theory at the Universite´ de Montreal.´ His main research interests lie in analytic and combinatorial number theory, and in solutions to Diophantine equations. He first became interested in distribution questions when studying primes and smooth numbers. His interest was piqued by Maier’s result that primes are not distributed in short intervals as well as the Gauss–Cramer´ model had suggested. Roger Heath-Brown was an undergraduate in Cambridge University, where he later did his doctorate under Alan Baker. He moved to Oxford in 1979,

xiii xiv BIOGRAPHICAL SKETCHES OF THE LECTURERS and became Professor of in 1999. His research has cov- ered a wide range of topics in analytic number theory, including the zeta- function, the distribution of primes, applications of sieves, the circle method, and exponential sums. Most recently he has been particularly interested in the application of analytic methods to the study of Diophantine Geometry. Elon Lindenstrauss studied mathematics both as an undergraduate and a grad- uate student at the Hebrew University in Jerusalem, obtaining his Ph.D. under the guidance of Benjamin Weiss. His first postgraduate position was at the Institute for Advanced Study in Princeton where he learnt about arithmetic quantum unique ergodicity and other arithmetic questions from Peter Sar- nak. He is currently Professor of Mathematics at Princeton University, and was a Clay Research Fellow for two years. His research interests include ergodic theory, dynamical systems, automorphic forms and number theory and particularly the interplay between these fields. Jens Marklof is Professor of Mathematical Physics at the University of Bris- tol. He was educated at the Universities of Hamburg, Princeton and Ulm, where he received his Ph.D. in 1997. Before taking up a lectureship at Bristol in 1999, Marklof held post-doctoral positions at Hewlett-Packard, Cambridge University, Universite´ Paris-Sud and IHES.´ His main interests are problems at the interface of dynamical systems, number theory and quantum mechanics. In 2004 he received the Marie Curie Award and the Philip Leverhulme Prize for his work on the spectral statistics of integrable quantum systems. Ze´ev Rudnick received his Ph.D. at in 1990. He was a Szego assistant professor at Stanford and following that an assistant professor at Princeton University. Since 1995 he has been at . Cur- rently, his main research interests are in analytic number theory and in quan- tum chaos. K. Soundararajan received his undergraduate degree from the University of Michigan and was a doctoral student of Peter Sarnak at Princeton Univer- sity. He held a fellowship from the American Institute of Mathematics, and was a postdoc at the Institute for Advanced Study, before moving back to the University of Michigan. From the Fall of 2006 he will be Professor of Mathematics at Stanford University. His main interests are in L-functions and multiplicative number theory, harmonic analysis, and combinatorial number theory. Yuri Tschinkel was educated at and M.I.T., and is currently (the Gauss) Chair of Pure Mathematics at the University of Gottingen,¨ and is Professor of Mathematics at the Courant Institute, NYU. His research interests lie in arithmetic algebraic geometry and analytic num- ber theory. In particular, he studies the distribution of rational and integral BIOGRAPHICAL SKETCHES OF THE LECTURERS xv points on higher-dimensional algebraic varieties, over number fields and func- tion fields. Emmanuel Ullmo is a professor at Universite´ Paris-Sud (Orsay). His main research interests lie in arithmetic geometry (Arakelov theory, modular and automorphic forms, Shimura varieties, ergodic theory). He has worked on several problems of equidistribution of sequences of points or positive di- mensional subvarieties of an algebraic variety: points with small heights on abelian varieties, CM points and Hecke points on Shimura varieties, equidis- tribution of special subvarieties of Shimura varieties. Akshay Venkatesh was an undergraduate at the University of Western Aus- tralia and received his Ph.D. from Princeton University. He is presently an As- sociate Professor at NYU. His research has focussed on problems with an an- alytic flavor in number theory and automorphic forms. He became very inter- ested in ergodic theory because “they” kept proving results that he couldn’t!