Results 1  10
of
28
On the geometry and cohomology of some simple Shimura varieties
, 1999
"... This paper has twin aims. On the one hand we prove the local Langlands conjecture for GL n over a padic 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 ladic cohomology of the "simple" Shimura varieti ..."
Abstract

Cited by 204 (16 self)
 Add to MetaCart
This paper has twin aims. On the one hand we prove the local Langlands conjecture for GL n over a padic 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 ladic 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 nontrivial additive character # : K
Functorial desingularization of quasiexcellent schemes in characteristic zero: the nonembedded case
, 2009
"... 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 ∈ Xre ..."
Abstract

Cited by 16 (1 self)
 Add to MetaCart
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
On nonabelian LubinTate theory via vanishing cycles
"... We give a purely local proof, in the depth 0 case, of the result by HarrisTaylor which asserts that local Langlands correspondence for GLn realizes itself inside the vanishing cycle cohomology of the deformation space of formal OKmodules of height n. Our proof is given by establishing the direct ..."
Abstract

Cited by 13 (4 self)
 Add to MetaCart
We give a purely local proof, in the depth 0 case, of the result by HarrisTaylor which asserts that local Langlands correspondence for GLn realizes itself inside the vanishing cycle cohomology of the deformation space of formal OKmodules of height n. Our proof is given by establishing the direct geometric link with the DeligneLusztig theory for GLn(Fq).
A trace formula for rigid varieties, and motivic Weil generating series for formal schemes
, 2009
"... 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 pseudofinite type over a complete disc ..."
Abstract

Cited by 8 (5 self)
 Add to MetaCart
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 pseudofinite type over a complete discrete valuation ring R, and we introduce the Weil generating series of a regular formal Rscheme X of pseudofinite type, via the construction of a GelfandLeray 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 socalled analytic Milnor fiber of f at x; we show that it completely determines the formal germ of f at x. 1.
The Bogomolov conjecture for totally degenerate abelian varietieties
"... 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 ..."
Abstract

Cited by 8 (3 self)
 Add to MetaCart
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
The motivic Serre invariant, ramification, and the analytic Milnor fiber
 Invent. Math
"... Let us recall the classical definition of a padic zeta function, as it was given by Igusa [29]. A survey of the theory of padic zeta functions can be found in Denef’s Bourbaki report [19]. ..."
Abstract

Cited by 8 (6 self)
 Add to MetaCart
Let us recall the classical definition of a padic zeta function, as it was given by Igusa [29]. A survey of the theory of padic zeta functions can be found in Denef’s Bourbaki report [19].
Vizio, Continuity of the radius of convergence of padic differential equations on berkovich spaces
, 2007
"... analytic spaces ..."
A trace formula for varieties over a discretely valued field
, 2008
"... We study the motivic Serre invariant of a smoothly bounded algebraic or rigid variety X over a complete discretely valued field K with perfect residue field k. If K has characteristic zero, we extend the definition to arbitrary Kvarieties using Bittner’s presentation of the Grothendieck ring and a ..."
Abstract

Cited by 4 (2 self)
 Add to MetaCart
We study the motivic Serre invariant of a smoothly bounded algebraic or rigid variety X over a complete discretely valued field K with perfect residue field k. If K has characteristic zero, we extend the definition to arbitrary Kvarieties using Bittner’s presentation of the Grothendieck ring and a process of Néron smoothening of pairs of varieties. The motivic Serre invariant can be considered as a measure for the set of unramified points on X. Under certain tameness conditions, it admits a cohomological interpretation by means of a trace formula. In the curve case, we use T. Saito’s geometric criterion for cohomological tameness to obtain more detailed results. We discuss some applications to WeilChâtelet groups, Chow motives, and the structure of the Grothendieck ring of varieties.