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

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

26 May 2020

## Journal:

Compositio Mathematica

## Last Updated:

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

6

156

## DOI:

10.1112/S0010437X20007149

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].

1088180

Submitted

Journal Article