Analytic methods in arithmetic geometry

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