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Shimura varieties at level
$\unicode[STIX]{x1D6E4}_{1}(p^{\infty })$ and Galois representations
Published online by Cambridge University Press: 26 May 2020
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].
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- Research Article
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- Copyright
- © The Authors 2020
Footnotes
Current address: Mathematical Institute, University of Oxford, Oxford OX2 6GG, UK email [email protected]
Current address: Department of Mathematics, University of California, Los Angeles, Box 95155, Los Angeles, CA 90095, USA email [email protected]
Current address: Department of Mathematics, University of Chicago, 5734 S. University Ave., Chicago, IL 60637, USA email [email protected]
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.
References
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