Shimura varieties at level $Γ_1(p^\infty)$ and Galois representations
File(s)Vanishingtheorems.pdf (879.68 KB)
Accepted version
Author(s)
Type
Journal Article
Abstract
We show that the compactly supported cohomology of certain $\mathrm{U}(n,n)$
or $\mathrm{Sp}(2n)$-Shimura varieties with $\Gamma_1(p^\infty)$-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 $\mathrm{GL}_n/F$. 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 and
Newton-Thorne.
or $\mathrm{Sp}(2n)$-Shimura varieties with $\Gamma_1(p^\infty)$-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 $\mathrm{GL}_n/F$. 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 and
Newton-Thorne.
Date Issued
2020-06
Date Acceptance
2020-02-06
Citation
Compositio Mathematica, 2020, 156 (6), pp.1152-1230
ISSN
0010-437X
Publisher
Foundation Compositio Mathematica
Start Page
1152
End Page
1230
Journal / Book Title
Compositio Mathematica
Volume
156
Issue
6
Copyright Statement
© The Authors 2020. This paper has been accepted for publication and will appear in a revised form, subsequent to peer-review and/or editorial input by Cambridge University Press.
Sponsor
The Royal Society
Identifier
https://www.cambridge.org/core/journals/compositio-mathematica/article/shimura-varieties-at-level-unicodestixx1d6e41pinfty-and-galois-representations/C3B16C7064EF287F66E0796B7CEBB591
Grant Number
UF150307
Subjects
math.NT
math.NT
math.AG
math.RT
Notes
v2: major revision to improve exposition, results are the same
Publication Status
Published
Date Publish Online
2020-05-26