On a conjecture of Jacquet

TL;DR

This paper proves Jacquet's conjecture on the nonvanishing of the triple product L-function at the central point for automorphic representations over any number field, linking it to quaternion algebra periods.

Contribution

It generalizes previous special case results to the full generality, establishing the conjecture for all number fields using advanced tools from automorphic forms and representation theory.

Findings

Proved the nonvanishing criterion for the triple product L-function in full generality.

Connected nonvanishing of L-values to quaternion algebra period integrals.

Extended previous results to arbitrary number fields with new techniques.

Abstract

In this note, we prove in full generality a conjecture of Jacquet concerning the nonvanishing of the triple product L-function at the central point. Let $\kay$ be a number field and let $\pi_i$, $i=1$, 2, 3 be cuspidal automorphic representations of $GL_2(\A)$ such that the product of their central characters is trivial. Then the central value $L(\frac12,\pi_1\otimes\pi_2\otimes\pi_3)$ of the triple product L--function is nonzero if and only if there exists a quaternion algebra $B$ over $\kay$ and automorphic forms $f_i^B\in \pi_i^B$, such that the integral of the product $f_1^B f_2^B f_3^B$ over the diagonal $Z(\Bbb A) B^\times(\kay) B^\times(\Bbb A)$ is nonzero, where $\pi_i^B$ is the representation of $B^\times(\A)$ corresponding to $\pi_i$. In a previous paper, we proved this conjecture in the special case where $\kay=\Q$ and the $\pi_i$'s correspond to a triple of holomorphic newforms. Recent improvement on the Ramanujan bound due to Kim and Shahidi, results about the local L-factors due to Ikeda and Ramakrishnan, results of Chen-bo Zhu and Sahi about invariant distributions and degenerate principal series in the complex case, and an extension of the Siegel--Weil formula to similitude groups allow us to carry over our method to the general case.