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Shimura varieties at level $\unicode[STIX]{x1D6E4}_{1}(p^{\infty })$ and Galois representations

Part of: Discontinuous groups and automorphic forms Arithmetic problems. Diophantine geometry Arithmetic algebraic geometry

Published online by Cambridge University Press:  26 May 2020

Ana Caraiani ,
Daniel R. Gulotta ,
Chi-Yun Hsu ,
Christian Johansson ,
Lucia Mocz ,
Emanuel Reinecke  and
Sheng-Chi Shih
Ana Caraiani
Affiliation:
Department of Mathematics, Imperial College London, LondonSW7 2AZ, UK email caraiani.ana@gmail.com
Daniel R. Gulotta
Affiliation:
Department of Mathematics, Columbia University, 2990 Broadway, New York, NY10027, USA
Chi-Yun Hsu
Affiliation:
Department of Mathematics, Harvard University, 1 Oxford Street, Cambridge, MA02138, USA
Christian Johansson
Affiliation:
Department of Mathematical Sciences, Chalmers University of Technology and the University of Gothenburg, SE-412 96Gothenburg, Sweden email chrjohv@chalmers.se
Lucia Mocz
Affiliation:
Mathematisches Institut der Universität Bonn, Endenicher Allee 60, 53115Bonn, Germany
Emanuel Reinecke
Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, MI48109, USA email reinec@umich.edu
Sheng-Chi Shih
Affiliation:
University of Lille, CNRS, UMR 8524 – Laboratoire Paul Painlevé, 59000Lille, France email sheng-chi.shih@univ-lille.fr
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Abstract

We show that the compactly supported cohomology of certain $\text{U}(n,n)$- or $\text{Sp}(2n)$-Shimura varieties with $\unicode[STIX]{x1D6E4}_{1}(p^{\infty })$-level vanishes above the middle degree. The only assumption is that we work over a CM field $F$ in which the prime $p$ splits completely. We also give an application to Galois representations for torsion in the cohomology of the locally symmetric spaces for $\text{GL}_{n}/F$. More precisely, we use the vanishing result for Shimura varieties to eliminate the nilpotent ideal in the construction of these Galois representations. This strengthens recent results of Scholze [On torsion in the cohomology of locally symmetric varieties, Ann. of Math. (2) 182 (2015), 945–1066; MR 3418533] and Newton–Thorne [Torsion Galois representations over CM fields and Hecke algebras in the derived category, Forum Math. Sigma 4 (2016), e21; MR 3528275].

Keywords

locally symmetric spacesperfectoid spacesGalois representations

MSC classification

Primary: 11F75: Cohomology of arithmetic groups 11F80: Galois representations 11G18: Arithmetic aspects of modular and Shimura varieties 14G22: Rigid analytic geometry
Type
Research Article
Information
Compositio Mathematica , Volume 156 , Issue 6 , June 2020 , pp. 1152 - 1230
Copyright
© The Authors 2020

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Footnotes

1

Current address: Mathematical Institute, University of Oxford, Oxford OX2 6GG, UK email Daniel.Gulotta@maths.ox.ac.uk

2

Current address: Department of Mathematics, University of California, Los Angeles, Box 95155, Los Angeles, CA 90095, USA email cyhsu@math.ucla.edu

3

Current address: Department of Mathematics, University of Chicago, 5734 S. University Ave., Chicago, IL 60637, USA email lmocz@math.uchicago.edu

A.C. was supported in part by a Royal Society University Research Fellowship and by ERC Starting Grant 804176. C.-Y.H. is partially supported by a Government Scholarship to Study Abroad from Taiwan. C.J. was supported in part by the Herchel Smith Foundation. E.R. was partially supported by NSF Grant No. DMS-1501461.

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