On torsion in the cohomology of locally symmetric varieties
type: Item
instance of: scholarly article
title: On torsion in the cohomology of locally symmetric varieties
zbMATH Open document ID: 1345.14031
author: Peter Scholze
published in: Annals of Mathematics. Second Series
publication date: 2015-11-30T00:00:00Z
full work available at URL: https://arxiv.org/abs/1306.2070
review text: The paper constructs Galois representations associated to Hecke eigenclasses in the torsion cohomology of \(\mathrm{GL}_n\). This holds over a field which is either totally real or CM, assuming conjectures of \textit{J. Arthur} [The endoscopic classification of representations. Orthogonal and symplectic groups. Providence, RI: American Mathematical Society (AMS) (2013; Zbl 1310.22014)], and without these for CM fields containing an imaginary quadratic field. For the proof the classes are realised in the boundary of Siegel spaces. For these an innovative new technique is used: The infinite cover defined by level-\(p\)-structures of all orders is a perfectoid space which extends to the minimal compactification and admits a Hodge-Tate period map to the dual compact space. Its singular cohomology is described by coherent cohomology, and Hecke eigenclasses can be lifted to classes which are not torsion. For these the machinery of the trace formula allows to construct Galois representations.
Mathematics Subject Classification ID: 14G20
Mathematics Subject Classification ID: 11F46
Mathematics Subject Classification ID: 11S37
Mathematics Subject Classification ID: 14G35
Mathematics Subject Classification ID: 11G18
Mathematics Subject Classification ID: 22E50
zbMATH DE Number: 6514750
zbMATH Keywords: automorphic forms
zbMATH Keywords: Galois representations
zbMATH Keywords: Shimura varieties
MaRDI profile type: Publication
Wikidata QID: Q90681723
OpenAlex ID: W2964112405
arXiv ID: 1306.2070
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On torsion in the cohomology of locally symmetric varieties (DOI)
DOI: 10.4007/ANNALS.2015.182.3.3

On torsion in the cohomology of locally symmetric varieties (reviewed by)
reviewed by: Gerd Faltings