Adjoint L-functions for GL(3) and U(2,1)

TL;DR

This paper proves the holomorphy of the finite part of the adjoint L-function for cuspidal automorphic representations of GL(3) over number fields, extending previous results and exploring related cases for unitary groups.

Contribution

It improves the understanding of the holomorphy of the adjoint L-function, including ramified places, and relates poles to endoscopy and base change for unitary groups.

Findings

Finite part of the adjoint L-function is holomorphic for Re(s) ≥ 1/2.

Extended results to twisted adjoint L-functions of GL(3) and quasisplit unitary groups.

Clarified the relationship between poles, endoscopy, and base change in unitary groups.

Abstract

We show that the finite part of the adjoint $L$ function (including contributions from all nonarchimedean places, including ramified places) is holomorphic in $\Re(s) \ge 1/2$ for a cuspidal automorphic representation of $GL_3$ over a number field. This improves the main result of [H16]. We obtain more general results for twisted adjoint $L$ functions of both $GL_3$ and quasisplit unitary groups. For unitary groups, we explicate the relationship between poles of twisted adjoint $L$ functions, endoscopy, and the structure of the stable base change lifting.