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

TL;DR

This paper proves the Equivariant Tamagawa Number Conjecture for modular motives with coefficients in Hecke algebras, using innovative methods combining Euler systems and Taylor-Wiles systems, and shows its compatibility with specialization.

Contribution

It introduces a novel approach combining Euler systems and Taylor-Wiles systems to prove the conjecture for modular motives with Hecke algebra coefficients.

Findings

Proved the conjecture under mild residual representation hypotheses.

Established compatibility of the conjecture with specialization.

Developed new methods combining Euler systems and Taylor-Wiles systems.

Abstract

Under mild hypotheses on the residual representation, we prove the Equivariant Tamagawa Number Conjecture for modular motives with coefficients in universal deformation rings and Hecke algebras using a novel combination of the methods of Euler systems and Taylor-Wiles systems. We also prove the compatibility of this conjecture with specialization.