Semisimplicity of \'etale cohomology of certain Shimura varieties

TL;DR

This paper proves the semisimplicity of specific parts of the étale cohomology of certain Shimura varieties, assuming the existence and properties of related automorphic Galois representations, advancing understanding in arithmetic geometry.

Contribution

It establishes semisimplicity results for étale cohomology of abelian-type Shimura varieties under new assumptions involving automorphic Galois representations.

Findings

Semisimplicity of étale cohomology parts proven under automorphic Galois assumptions

Combines Fayad-Nekovár criterion with Eichler-Shimura relations

Advances understanding of Shimura varieties in arithmetic geometry

Abstract

Building on work of Fayad and Nekov\'{a}\v{r}, we show that a certain part of the etale cohomology of some abelian-type Shimura varieties is semisimple, assuming the associated automorphic Galois representations exists, and satisfies some good properties. The proof combines an abstract semisimplicity criterion of Fayad-Nekov\'{a}\v{r} with the Eichler-Shimura relations.