Part of the Langlands functoriality conjecture asserts, roughly, that any reasonable operation on Galois representations admits an automorphic analogues. The operation of restricting a Galois representation of the absolute Galois group of a number field \(F\) to the absolute Galois group of an extension field \(E/F\) should induce a functorial transfer of automorphic representations known as base change.
This is a very basic operation on Galois representations, and it is perhaps surprising that the base change of an automorphic representation of \(\mathrm{GL}_n(\mathbb{A}_n)\) is only known to exist when \(E/F\) is solvable. This result was crucial in Langlands and Tunnel’s work on the Artin conjecture for \(2\)-dimensional solvable Galois representations, which in turn was used in the proof of Fermat’s last theorem.
One can essentially reduce to the case where \(\mathrm{Gal}(E/F)=\langle \sigma \rangle\) is cyclic. Following an idea of Saito and Shintani one compares a twisted trace formula isolating automorphic representations of \(\mathrm{GL}_n(\mathbb{A}_E)\) invariant under the generator of \(\mathrm{Gal}(E/F)=\langle \sigma \rangle\) and a trace formula encoding information about automorphic representations of \(\mathrm{GL}_n(\mathbb{A}_E)\).
It is natural to ask about the case where \(\mathrm{Gal}(E/F)\) is simple and nonabelian. In this setting, motivated by Langlands’ Beyond Endoscopy idea, Getz suggested constructing nonabelian trace formulae and comparing them to usual trace formulae. These are trace formulae isolating representation invariant under a pair of automorphisms. Getz constructed a nonabelian trace formula whose geometric side is reasonably explicit. The TRT will examine this expression and refine it with a view towards comparison with a usual trace formula.
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