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

TL;DR

This paper proves the vanishing of compactly supported cohomology above the middle degree for certain Shimura varieties at level (p^0), and applies this to improve the construction of Galois representations for torsion classes.

Contribution

It establishes a vanishing theorem for the cohomology of specific Shimura varieties at deep level and uses this to refine Galois representation constructions for torsion cohomology.

Findings

Vanishing of cohomology above middle degree for (p^0)-level Shimura varieties.

Elimination of the nilpotent ideal in Galois representation construction.

Strengthening of recent results by Scholze and Newton-Thorne.

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.