An approach to nonsolvable base change and descent

TL;DR

This paper explores conjectural trace identities that relate to the base change and descent of automorphic representations over nonsolvable extensions, with detailed analysis for the case n=2 and implications for the Artin conjecture.

Contribution

It introduces conjectural trace identities that are equivalent to base change and descent in nonsolvable cases, providing new insights and applications to the Artin conjecture.

Findings

Conjectural trace identities are proposed as equivalent to base change and descent.

Detailed treatment of the case n=2 with applications to icosahedral Galois representations.

Connections established between trace identities and the Artin conjecture.

Abstract

We present a collection of conjectural trace identities and explain why they are equivalent to base change and descent of automorphic representations of $\mathrm{GL}_n(\mathbb{A}_F)$ along nonsolvable extensions (under some simplifying hypotheses). The case $n=2$ is treated in more detail and applications towards the Artin conjecture for icosahedral Galois representations are given.