The Equivariant Tamagawa Number Conjectures for modular motives with coefficients in Hecke algebra

TL;DR

This paper investigates the equivariant properties of special values of L-functions for modular motives with Hecke algebra coefficients, establishing new connections with the Iwasawa Main Conjecture under broad ramification conditions.

Contribution

It proves that special L-values of modular motives exhibit equivariant properties aligned with Hecke actions and derives new implications for the Iwasawa Main Conjecture.

Findings

Confirmed equivariant properties of L-values for modular motives.

Deduced new properties of the Iwasawa Main Conjecture.

Extended understanding of Hecke algebra actions on motives.

Abstract

Modular motives have coefficients in Hecke algebras. According to the equivariant philosophy, special values of $L$-functions of eigencuspforms should therefore exhibit equivariant properties with respect to various Hecke actions. This manuscript shows that this is indeed the case at least under broad conditions on ramification and deduce from them new properties of the Iwasawa Main Conjecture for modular forms. This manuscript is dedicated to the memory of Jo\"el Bella\"iche.