A nonabelian trace formula

TL;DR

This paper develops a trace formula for automorphic representations of GL_2 over a nonabelian Galois extension, aiding the study of base change and descent in automorphic forms.

Contribution

It introduces a novel trace formula tailored for nonabelian Galois extensions, advancing the understanding of automorphic representation transfer.

Findings

Established a trace formula for nonabelian Galois extensions

Connected the spectral side to automorphic representations invariant under Galois conjugation

Provided tools for nonsolvable base change and descent analysis

Abstract

Let $E/F$ be an extension of number fields with $\mathrm{Gal}(E/F)$ simple and nonabelian. In [G] the first named author suggested an approach to nonsolvable base change and descent of automorphic representations of $\mathrm{GL}_2$ along such an extension. Motivated by this we prove a trace formula whose spectral side is a weighted sum over cuspidal automorphic representations of $\mathrm{GL}_2(\mathbb{A}_E)$ that are isomorphic to their $\mathrm{Gal}(E/F)$-conjugates.