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

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