Refined applications of Kato's Euler systems for modular forms

TL;DR

This paper advances the understanding of Kato's Euler systems for higher weight modular forms, providing new proofs and interpretations of key conjectures in Iwasawa theory and their implications for the Birch and Swinnerton-Dyer conjecture.

Contribution

It offers refined applications of Kato's Euler systems, including proofs of the Mazur--Tate conjecture and new insights into the Iwasawa main conjecture beyond existing divisibility results.

Findings

Proof of the Mazur--Tate conjecture on Fitting ideals

New interpretation of the Iwasawa main conjecture

Applications to Birch and Swinnerton-Dyer conjecture

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 via the non-triviality of Kato's Kolyvagin systems with structural applications. Some applications to Birch and Swinnerton-Dyer conjecture are also discussed.