On triple product L-functions

Abstract

Let $\pi ={\pi }_1\otimes {\pi }_2\otimes {\pi }_3$ be a unitary cuspidal automorphic representation of ${\mathrm{GL}}_3^3({\mathbb{A}}_F)$ where $F$ is a number field. Assume that $\pi$ is everywhere tempered. Under suitable local hypotheses, for a sufficiently large finite set of places $S$ of $F$ we prove that the triple product $L$-function $L^S(s,\pi ,{\otimes }^3)$ admits a meromorphic continuation to $\mathrm{Re}(s)>\frac{1}{2}$. We also give some information about the possible poles.

The main theorem relies on a soft method for isolating a cusp form in a family. Unfortunately there is a gap in the argument. The author is currently working on explicating the relevant Fourier transform so a more refined approach can be applied