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- W1610072717 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 $L(frac12,pi_1otimespi_2otimespi_3)$ of the triple product L--function is nonzero if and only if there exists a quaternion algebra $B$ over $kay$ and automorphic forms $f_i^Bin pi_i^B$, 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." @default.
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- W1610072717 title "On a conjecture of Jacquet" @default.
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