Shimura varieties at level $\Gamma_1(p^{\infty})$ and Galois representations

Author: 

Caraiani, A
Gulotta, D
Hsu, C
Johansson, H
Mocz, L
Reinecke, E
Shih, S

Publication Date: 

26 May 2020

Journal: 

Compositio Mathematica

Last Updated: 

2021-07-25T22:19:51.143+01:00

Issue: 

6

Volume: 

156

DOI: 

10.1112/S0010437X20007149

page: 

1152-1230

abstract: 

We show that the compactly supported cohomology of certain
U(n, n) or Sp(2n)-Shimura varieties with Γ1(p∞)-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 GLn/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 [Sch15] and Newton-Thorne [NT16].

Symplectic id: 

1088180

Submitted to ORA: 

Submitted

Publication Type: 

Journal Article