On a conjecture of Jacquet

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 $L(\frac{1}{2},{\pi }_1\otimes {\pi }_2\otimes {\pi }_3)$ 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.