Coherent cohomology and Galois representations
type: Item
instance of: scholarly article
title: Coherent cohomology and Galois representations
zbMATH Open document ID: 1357.14033
author: Vincent Pilloni
author: Benoît Stroh
published in: Annales Mathématiques du Québec
publication date: 2017-01-20T00:00:00Z
review text: The authors study the coherent cohomology of Shimura varieties of Hodge type, using the integral \(p\)-adic ``strange'' models of the minimal compactifications of these varieties introduced by \textit{P. Scholze} in his preprint [``On torsion in the cohomology of locally symmetric varieties'', Preprint, \url{arXiv:1306.2070}] and the existence, as soon as the level is ramified enough at \(p\), of certain torsion modular forms which commute with the action of the Hecke algebra of level prime to \(p\). These so-called ``Hodge-Tate classes'' advantageously constitute a linear system without base point of the suitably modified Hodge sheaf, which is ample over the strange model. A computation ``à la'' Chech then allows to represent the coherent cohomology classes by sections defined on open sets of non-vanishing of certain Hodge-Tate classes, which can be \(p\)-adically approximated, while respecting the Hecke action, by classical modular forms. The main result asserts that higher coherent cohomology classes are \(p\)-adic limits of cuspidal modular forms. In the first two sections of the paper, the authors take up again Scholze's construction of a formal model of the minimal compactification, first of Siegel varieties of level \(p^n\), then of Shimura varieties of Hodge type, in order to build a formal Hodge-Tate period map between minimal and toroidal compactifications. The point is that automorphic coherent sheaves are defined over the Kottwitz toroidal compactifications, whereas the proof of the main theorem requires to work with the strange models of the minimal compactification. Once the bridge is built, the authors can associate Galois representations to some automorphic representations appearing in the coherent cohomology of Shimura varieties. More precisely, let \(p\) be a prime number, \(G\) a reductive group over \(\mathbb Q\), \(K=K_pK^p\) an open compact subgroup of \(\mathrm{GL}(\mathbb A_f)\). Given a Shimura data for \(G\), suppose that the associated Shi…
reviewed by: Thong Nguyen Quang Do
Mathematics Subject Classification ID: 14G35
Mathematics Subject Classification ID: 11G18
Mathematics Subject Classification ID: 11F80
Mathematics Subject Classification ID: 11F46
zbMATH DE Number: 6676669
zbMATH Keywords: Shimura varieties
zbMATH Keywords: coherent cohomology
zbMATH Keywords: Galois representations
MaRDI profile type: Publication
full work available at URL: s40316-015-0056-0
OpenAlex ID: W2301332904
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DOI: 10.1007/S40316-015-0056-0
description: scientific article; zbMATH DE number 6676669