Equidistribution in Number Theory, 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 Sub-Series I. Life and Behavioural Sciences IOS Press II. Mathematics, Physics and Chemistry Springer III. Computer and Systems Science IOS Press IV. Earth and Environmental Sciences Springer The NATO Science Series continues the series of books published formerly as the NATO ASI Series. 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. Advanced Study Institutes are high-level tutorial courses offering in-depth study of latest advances in a field. Advanced Research Workshops are expert meetings aimed at critical assessment of a field, and identification of directions for future action. As a consequence of the restructuring of the NATO Science Programme in 1999, the NATO Science Series was re-organised to the four sub-series noted above. Please consult the following web sites for information on previous volumes published in the Series. http://www.nato.int/science http://www.springer.com http://www.iospress.nl Series II: Mathematics, Physics and Chemistry – Vol. 237 Equidistribution in Number Theory, An Introduction edited by Andrew Granville University of Montreal,´ QC, Canada and Zeév Rudnick Tel-Aviv University, Israel 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 Elon Lindenstrauss/ 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, Ergodic theory 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.