On triple product L-functions

TL;DR

This paper proves the meromorphic continuation of the triple product L-function for certain automorphic representations of GL(3) over number fields, under specific local conditions, and discusses potential poles.

Contribution

It establishes the meromorphic continuation of the triple product L-function for tempered automorphic representations of GL(3) with local hypotheses, extending previous results.

Findings

Meromorphic continuation of L-function for Re(s) > 1/2

Conditions under which poles may occur

Extension of known analytic properties of triple product L-functions

Abstract

Let $\pi=\pi_1 \otimes \pi_2 \otimes \pi_3$ be a unitary cuspidal automorphic representation of $\mathrm{GL}_3^3(\mathbb{A}_F)$ where $F$ is a number field. Assume that $\pi$ is everywhere tempered. Under suitable local hypotheses, for a sufficiently large finite set of places $S$ of $F$ we prove that the triple product $L$-function $L^S(s,\pi,\otimes^3)$ admits a meromorphic continuation to $\mathrm{Re}(s) >\tfrac{1}{2}$. We also give some information about the possible poles.