**Contemporary Mathematics**

Volume: 740;
2019;
248 pp;
Softcover

MSC: Primary 11; 20; 05; 19; 14;

**Print ISBN: 978-1-4704-3784-8
Product Code: CONM/740**

List Price: $117.00

AMS Member Price: $93.60

MAA Member Price: $105.30

**Electronic ISBN: 978-1-4704-5629-0
Product Code: CONM/740.E**

List Price: $117.00

AMS Member Price: $93.60

MAA Member Price: $105.30

# Analytic Methods in Arithmetic Geometry

Share this page *Edited by *
*Alina Bucur; David Zureick-Brown*

A co-publication of the AMS and Centre de Recherches Mathématiques

This volume contains the proceedings of the Arizona Winter School
2016, which was held from March 12–16, 2016, at The University of
Arizona, Tucson, AZ.

In the last decade or so, analytic methods have had great success
in answering questions in arithmetic geometry and number theory. The
School provided a unique opportunity to introduce graduate students to
analytic methods in arithmetic geometry.

The book contains four articles. Alina C. Cojocaru's article
introduces sieving techniques to study the group structure of points
of the reduction of an elliptic curve modulo a rational prime via its
division fields. Harald A. Helfgott's article provides an introduction
to the study of growth in groups of Lie type, with
\(\mathrm{SL}_2(\mathbb{F}_q)\) and some of its subgroups as the key
examples. The article by Étienne Fouvry, Emmanuel Kowalski, Philippe
Michel, and Will Sawin describes how a systematic use of the deep
methods from \(\ell\)-adic cohomology pioneered by Grothendieck and
Deligne and further developed by Katz and Laumon help make progress on
various classical questions from analytic number theory. The last
article, by Andrew V. Sutherland, introduces Sato-Tate groups and
explores their relationship with Galois representations, motivic
\(L\)-functions, and Mumford-Tate groups.

#### Readership

Graduate students and research mathematicians interested in arithmetic and algebraic geometry and analytic number theory.

# Table of Contents

## Analytic Methods in Arithmetic Geometry

- Cover Cover11
- Title page iii4
- Contents v6
- Preface vii8
- Primes, elliptic curves and cyclic groups 110
- 1. Introduction 211
- 2. Primes 211
- 3. Elliptic curves: generalities 716
- 4. Elliptic curves over \Q: group structure 918
- 5. Elliptic curves over \Q: division fields 1120
- 6. Elliptic curves over \Q: maximal Galois representations 1322
- 7. Elliptic curves over \Q: two-parameter families 1524
- 8. Elliptic curves over \Q: reductions modulo primes 1726
- 9. Cyclicity question: heuristics and upcoming challenges 2130
- 10. Cyclicity question: asymptotic 2534
- 11. Cyclicity question: lower bound 3140
- 12. Cyclicity question: average 3241
- 13. Primality of 𝑝+1-𝑎_{𝑝} 4150
- 14. Anomalous primes 4554
- 15. Global perspectives 4958
- 16. Final remarks 5160
- Acknowledgments 6473
- References 6473

- Growth and expansion in algebraic groups over finite fields 7180
- Lectures on applied ℓ-adic cohomology 113122
- 1. Introduction 113122
- 2. Examples of trace functions 114123
- 3. Trace functions and Galois representations 117126
- 4. Summing trace functions over \Fq 124133
- 5. Quasi-orthogonality relations 128137
- 6. Trace functions over short intervals 131140
- 7. Autocorrelation of trace functions; the automorphism group of a sheaf 135144
- 8. Trace functions vs. primes 137146
- 9. Bilinear sums of trace functions 139148
- 10. Trace functions vs. modular forms 141150
- 11. The ternary divisor function in arithmetic progressions to large moduli 147156
- 12. The geometric monodromy group and Sato-Tate laws 150159
- 13. Multicorrelation of trace functions 159168
- 14. Advanced completion methods: the 𝑞-van der Corput method 167176
- 15. Around Zhang’s theorem on bounded gaps between primes 172181
- 16. Advanced completions methods: the +𝑎𝑏 shift 181190
- Acknowledgements 192201
- References 192201

- Sato-Tate distributions 197206
- Back Cover Back Cover1258