A nonabelian trace formula

Jayce Robert Getz,Paul Edward Herman

doi:10.1186/s40687-015-0025-x

Jayce Robert Getz, Paul Edward Herman

Open Access

https://doi.org/10.1186/s40687-015-0025-x

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Abstract

Let E/F be an everywhere unramified extension of number fields with Gal(E/F) simple and nonabelian. In a recent paper, the first named author suggested an approach to nonsolvable base change and descent of automorphic representations of GL2 along such an extension. Motivated by this, we prove a trace formula whose spectral side is a weighted sum over cuspidal automorphic representations of GL 2 ( A E ) $\text {GL}_{2}(\mathbb {A}_{E})$ that are isomorphic to their Gal(E/F)-conjugates. The basic method, which is of interest in itself, is to use functions in a space isolated by Finis, Lapid, and Muller to build more variables into the trace formula. 2010 Mathematics subject classification: Primary 11F70, Secondary 11F66

Highlights

Let F be a number field and let H, G be connected reductive F-groups
One has a standard technique available to isolate representations that are isomorphic to their twists under an automorphism of G, namely the twisted trace formula, and one can compare this to a trace formula on H to establish the desired transfer; see [2] for the first set of examples in arbitrary rank and [1,22] for more general examples that represent the state of the art
If one could isolate the automorphic representations of a group G invariant under a noncyclic group of automorphisms, one could hope to establish new cases of Langlands functoriality

Summary

Introduction

Let F be a number field and let H, G be connected reductive F-groups. The Langlands functoriality conjecture predicts that given an L-map. If f ∈ Cc∞(GL2(AF )1), this formula is a special case of the Arthur-Selberg trace formula This is ([7], Theorem 1), with a caveat: the theorem in loc. (1) The result is better than stated above, for the terms in the trace formula define continuous seminorms on C(GL2(AF )1) (see [7], Theorem 1). Is to emphasize that one can substitute our test function into the trace formula and obtain a meaningful, absolutely convergent geometric expression for its trace using Finis, Lapid, and Müller’s extension of the Arthur-Selberg trace formula.

Building more variables into the trace formula

The Casselman-Shalika-Shintani formula

Test functions and transfer

The subgroup H