David Geraghty

A Family of Calabi-Yau Varieties and Potential Automorphy II

We prove potential modularity theorems for l-adic representations of any dimension. From these results we deduce the Sato-Tate conjecture for all elliptic curves with nonintegral j -invariant defined over a totally real field.

The Sato-Tate conjecture for Hilbert modular forms

We prove the Sato-Tate conjecture for Hilbert modular forms. More precisely, we prove the natural generalisation of the Sato-Tate conjecture for regular algebraic cuspidal automorphic representations of GL2(AF ), F a totally real field, which are not of CM type. The argument is based on the potential automorphy techniques developed by Taylor et. al., but makes use of automorphy lifting theorems over ramified fields, together with a “topological” argument with local deformation rings. In particular, we give a new proof of the conjecture for modular forms, which does not make use of potential automorphy theorems for non-ordinary n-dimensional Galois representations.

Modularity lifting beyond the Taylor–Wiles method

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

Potential automorphy and change of weight

\mathrm {GL}(n)

Serre weights for rank two unitary groups

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

l=p

\mathrm {GL}(2)