  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.
