The commutative algebra of congruence ideals and applications to number theory

Abstract

In his proof of Fermat's Last Theorem, Wiles deployed a commutative algebra technique, namely a numerical criterion for detecting isomorphisms of rings. In our recent work we pick up on Wiles' work and generalize the numerical criterion to ``higher codimension''. A critical ingredient is a notion of congruence module in higher codimension: this has turned out to be a key definition whose utility extends beyond the role it plays in the numerical criterion. In this paper we trace the origin of some of the ideas that led to our work, both in number theory and commutative algebra, and new directions that emerge from it. We introduce a related notion of a congruence ideal. When applied to deformation theory of Galois representations and Hecke algebras, which is the setting of Wiles's work on Fermat's Last Theorem, our work leads to the notion of congruence ideals for local deformation rings. This sheds light on the classically studied congruence ideals for global deformation rings and Hecke algebras. We outline applications of the commutative algebra we have developed to: (i) integral modularity lifting theorems in the context of weight one forms, and (ii) factorization formulas for congruence ideals of global deformation rings at augmentations induced by newforms in which local congruence ideals enter as the local terms. The latter leads to surprising relations between these local congruence ideals and local Tamagawa ideals of Bloch-Kato associated to the rank 3 adjoint motive of $f$.