2001/11/21 by Michael Harris, Harris, Michael, Stephen S. Kudla +1
Mathematics · #FOS: Mathematics #Number Theory (math.NT) #Representation Theory (math.RT) #math.NT #math.RT
paper · pdf · doi:10.48550/arxiv.math/0111238
arxiv created 2001/11/21 · arxiv updated 2009/11/30
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 πi, i=1, 2, 3 be cuspidal automorphic representations of GL2(\A) such that the product of their central characters is trivial. Then the central value L(\frac12,π1⊗π2⊗π3) of the triple product L--function is nonzero if and only if there exists a quaternion algebra B over \kay and automorphic forms fiB∈ πiB, such that the integral of the product f1B f2B f3B over the diagonal Z(\Bbb A) B^×(\kay) B^×(\Bbb A) is nonzero, where πiB is the representation of B^×(\A) corresponding to πi. In a previous paper, we proved this conjecture in the special case where \kay=\Q and the πi'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.