Torsion Galois representations over CM fields and Hecke algebras in the derived category

TL;DR

This paper develops a framework for constructing Galois representations over CM fields using Hecke algebras in the derived category, advancing the understanding of arithmetic manifolds and their symmetries.

Contribution

It introduces a novel method to build Galois representations with coefficients in derived Hecke algebras, extending previous approaches in the field.

Findings

Constructed endomorphism algebras in the derived category generated by Hecke operators

Established Galois representations with coefficients in these derived Hecke algebras

Sharpened recent results of P. Scholze using this new framework

Abstract

We construct algebras of endomorphisms in the derived category of the cohomology of arithmetic manifolds, which are generated by Hecke operators. We construct Galois representations with coefficients in these Hecke algebras and apply this technique to sharpen recent results of P. Scholze.