Let Mm be the formal scheme which represents the functor of deformations of a one-dimensional formal module over F̄p equipped with a level-m-structure. By work of Boyer (in mixed characterististic) and Harris and Taylor, the l-adic étale cohomology of the generic fibre Mm of Mm realizes simultaneously the local Langlands and Jacquet-Langlands correspondences. The proofs given so far use Drinfeld...

Inside the universal deformation space of a local Galois representation one has the set of deformations which are potentially semi-stable of given p-adic Hodge and Galois type. It turns out these points cut out a closed subspace of the deformation space. A deep conjecture due to Breuil-Mézard predicts that part of the structure of this space can be described in terms of the local Langlands corr...

One of the fundamental problems of local harmonic analysis is to classify the set Π(G) of equivalence classes of irreducible representations of G(F ). The problem separates naturally into two parts. The first is to establish the local Langlands correspondence. This conjecture of Langlands asserts that Π(G) is a disjoint union of finite subsets Πφ, indexed by (equivalence classes of) Langlands p...

Let F be a locally compact non-Archimedean field, and let B/F be a division algebra of dimension 4. The JacquetLanglands correspondence provides a bijection between smooth irreducible representations π′ of B× of dimension > 1 and irreducible cuspidal representations of GL2(F ). We present a new construction of this bijection in which the preservation of epsilon factors is automatic. This is don...

Let G be a unitary group over Q, associated to a CM-field F with totally real part F, with signature (1, 1) at all the archimedean places of F. For certain primes p, and X a Shimura variety associated to G with good reduction at p, we construct a stratification of the characteristic p fiber of X , whose closed strata are indexed by subsets S of the archimedean places of F. Certain closed strata...

Let G = Sp(2n,C) be a complex symplectic group. In the companion paper [math.AG/0601154], we introduced a (G × C × C)-variety N, which we call the exotic nilpotent cone. In this paper, we realize the Hecke algebra H of type C n with unequal parameters via equivariant algebraic K-theory in terms of the geometry of N. This enables us to establish a Deligne-Langlands type classification of “non-cr...

• The theory of admissible representations of GL(2,Qp) (or more generally, GL(2, F ) with F/Qp a finite extension). • The theory of automorphic representations of GL(2); in particular, the correspondence between Hecke eigenforms in the classical sense and automorphic representations. • The Jacquet-Langlands correspondence, relating automorphic forms on GL(2) with those on a division algebra. • ...