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 Etienne 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.
Author(s): Alina Bucur, David Zureick-Brown
Series: Centre de Recherches Mathemtiques Proceedings 740
Publisher: American Mathematical Society
Year: 2020
Language: English
Pages: 258
Cover......Page 1
Title page......Page 4
Contents......Page 6
Preface......Page 8
Primes, elliptic curves and cyclic groups......Page 10
2. Primes......Page 11
3. Elliptic curves: generalities......Page 16
4. Elliptic curves over \Q: group structure......Page 18
5. Elliptic curves over \Q: division fields......Page 20
6. Elliptic curves over \Q: maximal Galois representations......Page 22
7. Elliptic curves over \Q: two-parameter families......Page 24
8. Elliptic curves over \Q: reductions modulo primes......Page 26
9. Cyclicity question: heuristics and upcoming challenges......Page 30
10. Cyclicity question: asymptotic......Page 34
11. Cyclicity question: lower bound......Page 40
12. Cyclicity question: average......Page 41
13. Primality of ��+1-��_{��}......Page 50
14. Anomalous primes......Page 54
15. Global perspectives......Page 58
16. Final remarks......Page 60
References......Page 73
1. Introduction......Page 80
2. Elementary tools......Page 87
3. Growth in a solvable group......Page 90
4. Intersections with varieties......Page 98
5. Growth and diameter in \SL₂(��)......Page 109
6. Further perspectives and open problems......Page 114
References......Page 117
1. Introduction......Page 122
2. Examples of trace functions......Page 123
3. Trace functions and Galois representations......Page 126
4. Summing trace functions over \Fq......Page 133
5. Quasi-orthogonality relations......Page 137
6. Trace functions over short intervals......Page 140
7. Autocorrelation of trace functions; the automorphism group of a sheaf......Page 144
8. Trace functions vs. primes......Page 146
9. Bilinear sums of trace functions......Page 148
10. Trace functions vs. modular forms......Page 150
11. The ternary divisor function in arithmetic progressions to large moduli......Page 156
12. The geometric monodromy group and Sato-Tate laws......Page 159
13. Multicorrelation of trace functions......Page 168
14. Advanced completion methods: the ��-van der Corput method......Page 176
15. Around Zhang’s theorem on bounded gaps between primes......Page 181
16. Advanced completions methods: the +���� shift......Page 190
References......Page 201
1. An introduction to Sato-Tate distributions......Page 206
2. Equidistribution, L-functions, and the Sato-Tate conjecture for elliptic curves......Page 219
3. Sato-Tate groups......Page 230
4. Sato–Tate axioms and Galois endomorphism types......Page 240
References......Page 253
Back Cover......Page 258
