Refined applications of Kato's Euler systems for modular forms

Abstract

We discuss refined applications of Kato's Euler systems for modular forms of higher weight at good primes (with more emphasis on the non-ordinary ones) beyond the one-sided divisibility of the main conjecture and the finiteness of Selmer groups. These include a proof of the Mazur-Tate conjecture on Fitting ideals of Selmer groups over p-cyclotomic extensions and a new interpretation of the Iwasawa main conjecture with structural applications. Contents 1. Introduction 1 2. Galois and Iwasawa cohomologies 11 3. Mazur-Tate elements I: Betti construction 13 4. Construction of canonical Kato's Euler systems 17 5. Review of p-adic Hodge theory 22 6. Local Iwasawa theory and explicit reciprocity law à la Perrin-Riou 24 7. Mazur-Tate elements II: étale construction 29 8. Towards the "weak" main conjecture of Mazur-Tate 40 9. Towards the Iwasawa main conjecture for modular forms 45 10. Towards the µ = 0 conjecture of Coates-Sujatha 50 Appendix A. On the p-part of the BSD formula for elliptic curves of analytic rank one 53 Appendix B. Kato's Kolyvagin systems and Heegner point Kolyvagin systems 53 References 58