This paper has twin aims. On the one hand we prove the local Langlands conjecture for GL n over a p-adic field. On the other hand in many cases we are able to identify the action of the decomposition group at a prime of bad reduction on the l-adic cohomology of the "simple" Shimura varieties studied by Kottwitz in [Ko4]. These two problems go hand in hand. The local Langlands conjecture is one of those hydra like conjectures which seems to grow as it gets proved. However the generally accepted formulation seems to be the following (see [He2]). Let K be a finite extension of Q p . Fix a non-trivial additive character # : K

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For a Noetherian scheme X, let Xreg denote the regular locus of X. The scheme X is said to admit a resolution of singularities if there exists a blowup X ′ → X with center disjoint from Xreg and regular X ′. More generally, for a closed subscheme Z ⊂ X, let (X,Z)reg denote the set of points x ∈ Xreg

We give a purely local proof, in the depth 0 case, of the result by Harris-Taylor which asserts that local Langlands correspondence for GLn realizes itself inside the vanishing cycle cohomology of the deformation space of formal OK-modules of height n. Our proof is given by establishing the direct geometric link with the Deligne-Lusztig theory for GLn(Fq).

Let K = k(B) be a function field of an integral projective variety B over the algebraically closed field k such that B is regular in codimension 1. The set of

Let us recall the classical definition of a p-adic zeta function, as it was given by Igusa [29]. A survey of the theory of p-adic zeta functions can be found in Denef’s Bourbaki report [19].

analytic spaces

Abstract. We establish a trace formula for rigid varieties X over a complete discretely valued field, which relates the set of unramified points on X to the Galois action on its étale cohomology. We develop a theory of motivic integration for formal schemes of pseudo-finite type over a complete discrete valuation ring R, and we introduce the Weil generating series of a regular formal R-scheme X of pseudo-finite type, via the construction of a Gelfand-Leray form on its generic fiber. Our trace formula yields a cohomological interpretation of this Weil generating series. When Xis the formal completion of a morphism f from a smooth irreducible variety to the affine line, then its Weil generating series coincides (modulo normalization) with the motivic zeta function of f. When X is the formal completion of f at a closed point x of the special fiber f −1 (0), we obtain the local motivic zeta function of f at x. In the latter case, the generic fiber of X is the so-called analytic Milnor fiber of f at x; we show that it completely determines the formal germ of f at x. 1.

“This paper is dedicated to R. Thom who, by his enthusiasm, convinced me of the importance of the singularities and, by his patience, backed me up along my long research towards the blowups of the versal deformations, the geometries of these processes and the (strange) attractors tied up to these”. MSC (2000): 14B05, 14B07, 14E15, 32S30, 11R39, 11F70n-dimensional global correspondences of Langlands over singular schemes (II)

Let ϕ: Y → X be a morphism of finite type between schemes of locally finite type over a non-Archimedean field k, and let F be an étale constructible sheaf on Y. In [Ber2] we proved that if the torsion orders of F are prime to the characteristic of the residue field of k then the canonical homomorphisms (R q ϕ∗F) an → R q ϕ an ∗ F an are isomorphisms. In this paper we extend the above result to the class of sheaves F with torsion orders prime to the characteristic of k.