On the generic part of the cohomology of compact unitary Shimura varieties

@article{Caraiani2015OnTG,
  title={On the generic part of the cohomology of compact unitary Shimura varieties},
  author={Ana Caraiani and Peter Scholze},
  journal={arXiv: Number Theory},
  year={2015}
}
The goal of this paper is to show that the cohomology of compact unitary Shimura varieties is concentrated in the middle degree and torsion-free, after localizing at a maximal ideal of the Hecke algebra satisfying a suitable genericity assumption. Along the way, we establish various foundational results on the geometry of the Hodge-Tate period map. In particular, we compare the fibres of the Hodge-Tate period map with Igusa varieties. 
On the generic part of the cohomology of local and global Shimura varieties
Using the work of Fargues-Scholze, we prove a vanishing theorem for the generic unramified part of the cohomology of local Shimura varieties of general linear groups. This gives an alternative
ON THE SUPERCUSPIDAL COHOMOLOGY OF BASIC LOCAL SHIMURA VARIETIES
We prove that the supercuspidal cohomology of basic local Shimura varieties is concentrated in the middle degree, under a mild (and probably necessary) condition related to the local Langlands
Vanishing theorems for the mod p cohomology of some simple Shimura varieties
Abstract We show that the mod p cohomology of a simple Shimura variety treated in Harris-Taylor’s book vanishes outside a certain nontrivial range after localizing at any non-Eisenstein ideal of the
Torsion classes in the cohomology of KHT Shimura varieties
A particular case of Bergeron-Venkatesh's conjecture predicts that torsion classes in the cohomology of Shimura varieties are rather rare. According to this and for Kottwitz-Harris-Taylor type of
Automorphic congruences and torsion in the cohomology of a simple unitary Shimura variety
We first give a relative flexible process to construct torsion cohomology classes for Shimura varieties of Kottwitz-Harris-Taylor type with coefficient in a non too regular local system. We then
Deformation of rigid Galois representations and cohomology of certain quaternionic unitary Shimura variety
. In this article, we use deformation theory of Galois representations valued in the symplectic group of degree four to prove a freeness result for the cohomology of certain quaternionic unitary
Local-global compatibility of mod $p$ Langlands program for certain Shimura varieties
We generalize the local-global compatibility result in [21] to higher dimensional cases, by examining the relation between Scholze’s functor and cohomology of Kottwitz-Harris-Taylor type Shimura
On the ordinary Hecke orbit conjecture
We prove the ordinary Hecke orbit conjecture for Shimura varieties of abelian type at primes of good reduction. We make use of the global Serre-Tate coordinates of Chai as well as recent results of
Chow groups and L-derivatives of automorphic motives for unitary groups, II.
Abstract In this article, we improve our main results from [LL21] in two directions: First, we allow ramified places in the CM extension $E/F$ at which we consider representations that are spherical
On the \'etale cohomology of Hilbert modular varieties with torsion coefficients
We study the étale cohomology of Hilbert modular varieties, building on the methods introduced for unitary Shimura varieties in [CS17, CS19]. We obtain the analogous vanishing theorem: in the
...
...

References

SHOWING 1-10 OF 101 REFERENCES
On the rigid cohomology of certain Shimura varieties
We construct the compatible system of l-adic representations associated to a regular algebraic cuspidal automorphic representation of $$GL_n$$GLn over a CM (or totally real) field and check
Torsion in the Coherent Cohomology of Shimura Varieties and Galois Representations
We introduce a method for producing congruences between Hecke eigenclasses, possibly torsion, in the coherent cohomology of automorphic vector bundles on certain good reduction Shimura varieties. The
p-adic Hodge-theoretic properties of \'etale cohomology with mod p coefficients, and the cohomology of Shimura varieties
We show that the mod p cohomology of a smooth projective variety with semistable reduction over K, a finite extension of Qp, embeds into the reduction modulo p of a semistable Galois representation
On certain unitary group Shimura varieties
— In this paper, we study the local geometry at a prime p of a certain class of (PEL) type Shimura varieties. We begin by studing the Newton polygon stratification of the special fiber of a Shimura
Étale Cohomology of Rigid Analytic Varieties and Adic Spaces
Summary of the results on the etale cohomology of rigid analytic varieties - Adic spaces - The etale site of a rigid analytic variety and an adic space - Comparison theorems - Base change theorems -
On the cohomology of certain PEL-type Shimura varieties
In this article we study the local geometry at a prime p of PEL-type Shimura varieties for which there is a hyperspecial level subgroup. We consider the Newton polygon stratification of the special
The geometry of Newton strata in the reduction modulo p of Shimura varieties of PEL type
In this thesis we study the Newton stratification on the reduction of Shimura varieties of PEL type with hyperspecial level structure. The main result is a formula for the dimension of Newton strata
Mod points on Shimura varieties of abelian type
We show that the mod p points on a Shimura variety of abelian type with hyperspecial level, have the form predicted by the conjectures of Kottwitz and Langlands-Rapoport. Along the way we show that
Galois representations arising from some compact Shimura varieties
Our aim is to establish some new cases of the global Langlands correspondence for GLm. Along the way we obtain a new result on the description of the cohomology of some compact Shimura varieties. Let
...
...