We prove an automorphy lifting theorem for l-adic representations where we impose a new
condition at l, which we call" potentential diagonalizability." This result allows for" change of …

We show that abelian surfaces (and consequently curves of genus 2) over totally real fields
are potentially modular. As a consequence, we obtain the expected meromorphic …

We prove new automorphy lifting theorems for essentially conjugate self-dual Galois
representations into GLn. Existing theorems require that the residual representation have …

We prove the Breuil–Mézard conjecture for two-dimensional potentially Barsotti–Tate
representations of the absolute Galois group (up to the question of determining precise …

We generalize results of Clozel, Harris and Taylor by proving modularity lifting theorems for
ordinary l-adic Galois representations of any dimension of an imaginary CM or totally real …

We formulate a number of related generalisations of the weight part of Serre's conjecture to
the case of GLn over an arbitrary number field, motivated by the formalism of the Breuil …

We prove a variety of results on the existence of automorphic Galois representations lifting a
residual automorphic Galois representation. We prove a result on the structure of …

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 …

We study the weight part of (a generalisation of) Serre's conjecture for mod l Galois
representations associated to automorphic representations on rank two unitary groups for …

Let G be a split connected reductive algebraic group over Q p such that both G and its dual
group G ˆ have connected centers. Motivated by a hypothetical p-adic Langlands …