Poles of triple product $L$-functions involving monomial representations

TL;DR

This paper investigates the poles of triple product $L$-functions involving monomial automorphic representations, providing insights relevant to Langlands' beyond endoscopy program.

Contribution

It determines the order of poles of triple tensor product $L$-functions when one representation is monomial, connecting to Brauer theory and Langlands' beyond endoscopy.

Findings

Identifies conditions for poles of triple $L$-functions involving monomial representations.

Provides examples relevant to Langlands' beyond endoscopy.

Enhances understanding of automorphic $L$-functions in the monomial case.

Abstract

In this paper, we study the order of the pole of the triple tensor product $L$-functions $L(s,\pi_1\times\pi_2\times\pi_3,\otimes^3)$ for cuspidal automorphic representations $\pi_i$ of $\mathrm{GL}_{n_i}(\mathbb{A}_F)$ in the setting where one of the $\pi_i$ is a monomial representation. In the view of Brauer theory, this is a natural setting to consider. The results provided in this paper give crucial examples that can be used as a point of reference for Langlands' beyond endoscopy proposal.