Companions on Artin stacks

TL;DR

This paper extends the theory of $\u03bb'$-companions and related theorems from schemes to smooth Artin stacks, confirming Deligne's conjecture in this broader context and advancing the understanding of $$-adic sheaves on stacks.

Contribution

It generalizes Drinfeld's theorem on $$-companions from schemes to smooth Artin stacks and applies these results to coarse moduli spaces.

Findings

Extension of Drinfeld's theorem to smooth Artin stacks

Confirmation of Deligne's conjecture for coarse moduli spaces of stacks

Generalization of Frobenius eigenvalue and trace theorems to stacks

Abstract

Deligne's conjecture that $\ell$-adic sheaves on normal schemes over a finite field admit $\ell'$-companions was proved by L. Lafforgue in the case of curves and by Drinfeld in the case of smooth schemes. In this paper, we extend Drinfeld's theorem to smooth Artin stacks and deduce Deligne's conjecture for coarse moduli spaces of smooth Artin stacks. We also extend related theorems on Frobenius eigenvalues and traces to Artin stacks.