On quadratic distinction of automorphic sheaves

TL;DR

This paper establishes a geometric characterization of automorphic sheaves related to quadratic base change, linking nonvanishing cohomology to conjugate self-duality of local systems, applicable over any base field.

Contribution

It provides a geometric analogue of classical automorphic representation results, connecting cohomology of automorphic sheaves to conjugate self-duality and base change for unitary groups.

Findings

Nonvanishing cohomology corresponds to conjugate self-dual local systems.

Geometric setting extends classical results to arbitrary base fields.

Characterizes automorphic sheaves via period integrals and local system properties.

Abstract

We prove a geometric version of a classical result on the characterization of an irreducible cuspidal automorphic representation of $\mathrm{GL}_n(\mathbb{A}_E)$ being the base change of a stable cuspidal packet of the quasi-split unitary group associated to the quadratic extension $E/F$, via the nonvanishing of certain period integrals, called being distinguished. We show that certain cohomology of an automorphic sheaf of $\mathrm{GL}_{n,X'}$ is nonvanishing if and only if the corresponding local system $E$ on $X'$ is conjugate self-dual with respect to an \'{e}tale double cover $X'/X$ of curves, which directly relates to the base change from the associated unitary group. In particular, the geometric setting makes sense for any base field.