Explore the vast range of titles available.

The main result of this paper is the existence of Galois representations associated with the mod p (or mod pm) cohomology of the locally symmetric spaces for GLn over a totally real or CM field, proving conjectures of Ash and others. Following an old suggestion of Clozel, recently realized by Harris-Lan-Taylor-Thorne for characteristic 0 cohomology classes, one realizes the cohomology of the locally symmetric spaces for GLn as a boundary contribution of the cohomology of symplectic or unitary Shimura varieties, so that the key problem is to understand torsion in the cohomology of Shimura varieties. Thus, we prove new results on the p-adic geometry of Shimura varieties (of Hodge type). Namely, the Shimura varieties become perfectoid when passing to the inverse limit over all levels at p, and a new period map towards the flag variety exists on them, called the Hodge-Tate period map. It is roughly analogous to the embedding of the hermitian symmetric domain (which is roughly the inverse limit over all levels of the complex points of the Shimura variety) into its compact dual. The Hodge-Tate period map has several favorable properties, the most important being that it commutes with the Hecke operators away from p (for the trivial action of these Hecke operators on the flag variety), and that automorphic vector bundles come via pullback from the flag variety.

We prove that the Witt vector affine Grassmannian, which parametrizes W ( k )-lattices in W ( k ) [ 1 p ] n for a perfect field k of characteristic p , is representable by an ind-(perfect scheme) over k . This improves on previous results of Zhu by constructing a natural ample line bundle. Along the way, we establish various foundational results on perfect schemes, notably h -descent results for vector bundles.

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.

We extend our methods from Scholze (Invent. Math. 2012 , doi: 10.1007/s00222-012-0419-y ) to reprove the Local Langlands Correspondence for GL n over p -adic fields as well as the existence of ℓ -adic Galois representations attached to (most) regular algebraic conjugate self-dual cuspidal automorphic representations, for which we prove a local-global compatibility statement as in the book of Harris-Taylor (The Geometry and Cohomology of Some Simple Shimura Varieties, 2001 ). In contrast to the proofs of the Local Langlands Correspondence given by Henniart (Invent. Math. 139(2), 439–455, 2000 ), and Harris-Taylor (The Geometry and Cohomology of Some Simple Shimura Varieties, 2001 ), our proof completely by-passes the numerical Local Langlands Correspondence of Henniart (Ann. Sci. Éc. Norm. Super. 21(4), 497–544, 1988 ). Instead, we make use of a previous result from Scholze (Invent. Math. 2012 , doi: 10.1007/s00222-012-0419-y ) describing the inertia-invariant nearby cycles in certain regular situations.

The author would like to make some changes to the previously published article [1] by correcting two definitions.

In this article, we prove results about the cohomology of compact unitary group Shimura varieties at split places. In nonendoscopic cases, we are able to give a full description of the cohomology, after restricting to integral Hecke operators at pp on the automorphic side. We allow arbitrary ramification at pp; even the PEL data may be ramified. This gives a description of the semisimple local Hasse-Weil zeta function in these cases. We also treat cases of nontrivial endoscopy. For this purpose, we give a general stabilization of the expression given in the article http://dx.doi.org/ 10.1090/S0894-0347-2012-00753-X, following the stabilization given by Kottwitz. This introduces endoscopic transfers of the functions ϕτ,h\\phi _{\\tau ,h} introduced in the above article. We state a general conjecture relating these endoscopic transfers with Langlands parameters. We verify this conjecture in all cases of EL type and deduce new results about the endoscopic part of the cohomology of Shimura varieties. This allows us to simplify the construction of Galois representations attached to conjugate self-dual regular algebraic cuspidal automorphic representations of GLn\\mathrm {GL}_n.

We show how the Langlands-Kottwitz method can be used to determine the semisimple local factors of the Hasse-Weil zeta-function of certain Shimura varieties. This is made possible by a new result describing part of the nearby cycle sheaves in certain situations. In combination with a general base-change lemma in harmonic analysis, we use this to prove a conjecture of Haines and Kottwitz for these Shimura varieties.

We extend our methods from Scholze (Invent. Math. 2012 , doi:10.1007/s00222-012-0419-y) to reprove the Local Langlands Correspondence for GL^sub n^ over p-adic fields as well as the existence of -adic Galois representations attached to (most) regular algebraic conjugate self-dual cuspidal automorphic representations, for which we prove a local-global compatibility statement as in the book of Harris-Taylor (The Geometry and Cohomology of Some Simple Shimura Varieties, 2001 ). In contrast to the proofs of the Local Langlands Correspondence given by Henniart (Invent. Math. 139(2), 439-455, 2000 ), and Harris-Taylor (The Geometry and Cohomology of Some Simple Shimura Varieties, 2001 ), our proof completely by-passes the numerical Local Langlands Correspondence of Henniart (Ann. Sci. Éc. Norm. Super. 21(4), 497-544, 1988 ). Instead, we make use of a previous result from Scholze (Invent. Math. 2012 , doi:10.1007/s00222-012-0419-y) describing the inertia-invariant nearby cycles in certain regular situations.[PUBLICATION ABSTRACT]

We extend our methods from Scholze (Invent. Math. 2012, doi:10.1007/s00222-012-0419-y) to reprove the Local Langlands Correspondence for GL sub(n) over p-adic fields as well as the existence of -adic Galois representations attached to (most) regular algebraic conjugate self-dual cuspidal automorphic representations, for which we prove a local-global compatibility statement as in the book of Harris-Taylor (The Geometry and Cohomology of Some Simple Shimura Varieties, 2001). In contrast to the proofs of the Local Langlands Correspondence given by Henniart (Invent. Math. 139(2), 439-455, 2000), and Harris-Taylor (The Geometry and Cohomology of Some Simple Shimura Varieties, 2001), our proof completely by-passes the numerical Local Langlands Correspondence of Henniart (Ann. Sci. Ec. Norm. Super. 21(4), 497-544, 1988). Instead, we make use of a previous result from Scholze (Invent. Math. 2012, doi:10.1007/s00222-012-0419-y) describing the inertia-invariant nearby cycles in certain regular situations.