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We propose a formulation of the Equivariant Tamagawa Number Conjecture formodular motives with coefficients in universal deformation rings and Heckealgebras; something which seems to have been heretofore missing because thecomplexes of Galois cohomology required were not known to be perfect. We showthat the fundamental line of this conjecture satisfies the expectedcompatibility property at geometric points (more precisely at the pointssatisfying the Weight-Monodromy conjecture) and is compatible withlevel-lowering and level-raising. Combining these properties with the methodsof Euler and Taylor-Wiles systems, we prove a significant part of the ETNC withcoefficients in Hecke algebras for motives attached to modular forms and thefull conjecture when the main conjecture of Iwasawa theory for modular forms isknown to be true.

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