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

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.