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 article shows that this is indeed the case at least under broad conditions on ramification and deduces from them new properties of the Iwasawa main conjecture for modular forms.

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