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IntroductionIn this paper, building on work of Wiles [Wi] and of Taylor and Wiles [TW], we will prove the following two theorems (see §2.2). Theorem A. If E /Q is an elliptic curve, then E is modular. Theorem B. If ρ : Gal(Q/Q) → GL 2 (F 5 ) is an irreducible continuous representation with cyclotomic determinant, then ρ is modular.We will first remind the reader of the content of these results and then briefly outline the method of proof.If N is a positive integer, then we let Γ 1 (N ) denote the subgroup of SL 2 (Z) consisting of matrices that modulo N are of the form 1 * 0 1 .The quotient of the upper half plane by Γ 1 (N ), acting by fractional linear transformations, is the complex manifold associated to an affine algebraic curve Y 1 (N ) /C . This curve has a natural model Y 1 (N ) /Q , which for N > 3 is a fine moduli scheme for elliptic curves with a point of exact order N . We will let X 1 (N ) denote the smooth projective curve which contains Y 1 (N ) as a dense Zariski open subset.Recall that a cusp form of weight k ≥ 1 and level N ≥ 1 is a holomorphic function f on the upper half complex plane H such that• for all matrices Key words and phrases. Elliptic curve, Galois representation, modularity. The first author was supported by the CNRS. The second author was partially supported by a grant from the NSF. The third author was partially supported by a grant from the NSF and an AMS Centennial Fellowship, and was working at Rutgers University during much of the research. The fourth author was partially supported by a grant from the NSF and by the Miller Institute for Basic Science. The space S k (N ) of cusp forms of weight k and level N is a finite-dimensional complex vector space. If f ∈ S k (N ), then it has an expansionand we define the L-series of f to befor anywith c ≡ 0 mod N and d ≡ p mod N . The operators T p for p | N can be simultaneously diagonalised on the space S k (N ) and a simultaneous eigenvector is called an eigenform. If f is an eigenform, then the corresponding eigenvalues, a p (f ), are algebraic integers and we haveLet λ be a place of the algebraic closure of Q in C above a rational prime and let Q λ denote the algebraic closure of Q thought of as a Q algebra via λ. If f ∈ S k (N ) is an eigenform, then there is a unique continuous irreducible representation [DS]. Moreover ρ is odd in the sense that det ρ of complex conjugation is −1. Also, ρ f,λ is potentially semi-stable at in the sense of Fontaine. We can choose a conjugate of ρ f,λ which is valued in GL 2 (O Q λ ), and reducing modulo the maximal ideal and semi-simplifying yields a continuous representationwhich, up to isomorphism, does not depend on the choice of conjugate of ρ f,λ . Now suppose that ρ : G Q → GL 2 (Q ) is a continuous representation which is unramified outside finitely many primes and for which the restriction of ρ to a decomposition group at is potentially semi-stable in the sense of Fontaine. To ρ| Gal(Q /Q ) we can associate both a pair of Hodge-Tate numbers and a Weil-Deligne representation of the Weil g...

We extend the methods of Wiles and of Taylor and Wiles from GL 2 to higher rank unitary groups and establish the automorphy of suitable conjugate self-dual, regular (de Rham with distinct Hodge-Tate numbers), minimally ramified, l-adic lifts of certain automorphic mod l Galois representations of any dimension. We also make a conjecture about the structure of mod l automorphic forms on definite unitary groups, which would generalise a lemma of Ihara for GL 2 . Following Wiles' method we show that this conjecture implies that our automorphy lifting theorem could be extended to cover lifts that are not minimally ramified.

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.

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