Modularity Lifting beyond the Taylor-Wiles Method

TL;DR

This paper extends modularity lifting theorems to broader contexts, removing previous restrictions related to cohomology degrees and automorphic form weights, and applies to general number fields and Galois representations.

Contribution

It introduces new modularity lifting theorems applicable beyond traditional settings, including cases over general number fields and torsion cohomology classes, with some results unconditional.

Findings

Potential automorphy of elliptic curves over arbitrary number fields.

Identification of deformation rings with Hecke algebras in GL(2) over Q.

Complete solution for multiplicity of modular representations in Jacobians for odd p.

Abstract

We prove new modularity lifting theorems for p-adic Galois representations in situations where the methods of Wiles and Taylor--Wiles do not apply. Previous generalizations of these methods have been restricted to situations where the automorphic forms in question contribute to a single degree of cohomology. In practice, this imposes several restrictions -- one must be in a Shimura variety setting and the automorphic forms must be of regular weight at infinity. In this paper, we essentially show how to remove these restrictions. Our most general result is a modularity lifting theorem which, on the automorphic side applies to automorphic forms on the group GL(n) over a general number field; it is contingent on a conjecture which, in particular, predicts the existence of Galois representations associated to torsion classes in the cohomology of the associated locally symmetric space. We show that if this conjecture holds, then our main theorem implies the following: if E is an elliptic curve over an arbitrary number field, then E is potentially automorphic and satisfies the Sato--Tate conjecture. In addition, we also prove some unconditional results. For example, in the setting of GL(2) over Q, we identify certain minimal global deformation rings with the Hecke algebras acting on spaces of p-adic Katz modular forms of weight one. Such algebras may well contain p-torsion. Moreover, we also completely solved the problem (for p odd) of determining the multiplicity of an irreducible modular representation rhobar in the Jacobian J_1(N), where N is the minimal level such that rhobar arises in weight two.