On a conjecture of Jacquet
| dc.creator | Harris, Michael | |
| dc.creator | Kudla, Stephen S. | |
| dc.date | 2001-11-21 | |
| dc.date.accessioned | 2026-07-07T04:44:43Z | |
| dc.date.available | 2026-07-07T04:44:43Z | |
| dc.description | 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 $π_i$, $i=1$, 2, 3 be cuspidal automorphic representations of $GL_2(\A)$ such that the product of their central characters is trivial. Then the central value $L(\frac12,π_1\otimesπ_2\otimesπ_3)$ of the triple product L--function is nonzero if and only if there exists a quaternion algebra $B$ over $\kay$ and automorphic forms $f_i^B\in π_i^B$, 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 $π_i^B$ is the representation of $B^\times(\A)$ corresponding to $π_i$. In a previous paper, we proved this conjecture in the special case where $\kay=\Q$ and the $π_i$'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. | |
| dc.identifier | https://arxiv.org/abs/math/0111238 | |
| dc.identifier | http://arxiv.org/abs/math/0111238 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/62705 | |
| dc.subject | Number Theory | |
| dc.subject | Representation Theory | |
| dc.title | On a conjecture of Jacquet | |
| dc.type | text |