We prove an automorphy lifting theorem for l-adic representations where we impose a new condition at l, which we call 'potential diagonalizability'. This result allows for 'change of weight' and seems to be substantially more flexible than previous theorems along the same lines. We derive several applications. For instance we show that any irreducible, totally odd, essentially self-dual, regular, weakly compatible system of l-adic representations of the absolute Galois group of a totally real field is potentially automorphic, and hence is pure and its L-function has meromorphic continuation to the whole complex plane and satisfies the expected functional equation.

We prove new potential modularity theorems for n-dimensional essentially self-dual ladic representations of the absolute Galois group of a totally real field. Most notably, in the ordinary case we prove quite a general result. Our results suffice to show that all the symmetric powers of any non-CM, holomorphic, cuspidal, elliptic modular newform of weight greater than one are potentially cuspidal automorphic. This in turns proves the Sato-Tate conjecture for such forms. (In passing we also note that the Sato-Tate conjecture can now be proved for any elliptic curve over a totally real field.) 2010 Mathematics Subject Classification: 11F80, 11F11, 11R45.

We use the patching method of Taylor-Wiles and Kisin to construct a candidate for the p-adic local Langlands correspondence for GL n (F ), F a finite extension of Q p . We use our construction to prove many new cases of the Breuil-Schneider conjecture.

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 1. 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 ρ in the Jacobian J1(N ), where N is the minimal level such that ρ arises in weight two.

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 odd primes l. We propose a conjectural set of Serre weights, agreeing with all conjectures in the literature, and under a mild assumption on the image of the mod l Galois representation we are able to show that any modular representation is modular of each conjectured weight. We make no assumptions on the ramification or inertial degrees of l. Our main innovation is to make use of the lifting techniques introduced in [BLGG11], [BLGG10] and [BLGGT10].