Modularity lifting beyond the Taylor–Wiles method

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 $$\mathrm {GL}(n)$$ 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 $$\mathrm {GL}(2)$$ over $$\mathbf {Q}$$ , we identify certain minimal global deformation rings with the Hecke algebras acting on spaces of p-adic Katz modular forms of weight 1. Such algebras may well contain p-torsion. Moreover, we also completely solve the problem (for p odd) of determining the multiplicity of an irreducible modular representation $$\overline{\rho }$$ in the Jacobian $$J_1(N)$$ , where N is the minimal level such that $$\overline{\rho }$$ arises in weight two.