Subconvexity bounds for triple $L$-functions and representation theory

Abstract

We describe a new method to estimate the trilinear period on automorphic representations of $\operatorname{PGL}_2(\mathbb{R})$. Such a period gives rise to a special value of the triple $L$-function. We prove a bound for the triple period which amounts to a subconvexity bound for the corresponding special value of the triple $L$-function. Our method is based on the study of the analytic structure of the corresponding unique trilinear functional on unitary representations of $\operatorname{PGL}_2(\mathbb{R})$.