Kolyvagin systems

Introduction......Page 5
0.1. Selmer sheaves and Kolyvagin systems......Page 7
0.2. Resemblance to the leading term of an L-function......Page 8
0.3. Applications......Page 9
0.4. Layout of the paper......Page 10
0.6. Acknowledgments......Page 11
1.1. Local conditions......Page 13
1.2. The finite/singular homomorphism......Page 16
1.3. Local duality......Page 17
2.1. Selmer modules......Page 19
2.3. Global duality......Page 22
3.1. Kolyvagin systems......Page 25
3.2. Euler systems and Kolyvagin systems......Page 29
3.3. Simplicial sheaves and Selmer groups......Page 30
3.4. Sheaves and monodromy......Page 32
3.5. Hypotheses on T, F, and P......Page 33
3.6. Choosing useful primes......Page 36
3.7. Some remarks about hypothesis (H.6)......Page 39
4.1. The core Selmer module......Page 41
4.2. Kolyvagin systems and the core rank......Page 46
4.3. The sheaf of stub Selmer modules......Page 47
4.4. Kolyvagin systems and the stub Selmer sheaf......Page 51
4.5. Kolyvagin systems over principal artinian rings......Page 54
5.1. Kolyvagin systems over a field......Page 61
5.2. Kolyvagin systems over a discrete valuation ring......Page 62
5.3. Kolyvagin systems over......Page 66
6.1. The multiplicative group......Page 75
6.2. Elliptic curves......Page 79
6.3. The multiplicative group, revisited......Page 82
Appendix A. Proof of Theorem 3.2.4......Page 85
Appendix B. Proof of Theorem 4.3.3, by Benjamin Howard......Page 95
Bibliography......Page 100