Landau-Siegel Zeros of Triple Product L-functions

TL;DR

This paper establishes zero-free regions for triple product L-functions associated with automorphic representations over number fields, ruling out Landau-Siegel zeros in general cases and identifying conditions for their possible existence.

Contribution

It proves the absence of Landau-Siegel zeros for certain triple product L-functions in the general type case and characterizes when such zeros could occur in non-general type cases.

Findings

Proved zero-free regions for triple product L-functions of general type.

Identified conditions for potential Landau-Siegel zeros in non-general type cases.

Extended understanding of zeros of automorphic L-functions over number fields.

Abstract

Let $F$ be a number field. Let $\pi_1,\pi_2$ be cuspidal automorphic representations of $GL_2(\mathbb{A}_F)$, and let $\pi$ be a cuspidal automorphic representation of either $GL_2(\mathbb{A}_F)$ or $GL_3(\mathbb{A}_F)$. When $(\pi_1,\pi_2,\pi)$ is of general type, we show that the triple product $L$-function $L(s,\pi_1 \times \pi_2 \times \pi)$ on either $GL(2) \times GL(2) \times GL(2)$ or $GL(2) \times GL(2) \times GL(3)$ has a standard zero-free region with no exceptional Landau-Siegel zero. Moreover, when $(\pi_1,\pi_2,\pi)$ is not of general type, we give precise conditions when $L(s,\pi_1 \times \pi_2 \times \pi)$ could possibly have exceptional Landau-Siegel zeros.