The Weyl bound for triple product L-functions

Abstract

Let $\pi_1, \pi_2, \pi_3$ be three cuspidal automorphic representations for the group ${\rm SL}(2, \Bbb{Z})$, where $\pi_1$ and $\pi_2$ are fixed and $\pi_3$ has large conductor. We prove a subconvex bound for $L(1/2, \pi_1 \otimes \pi_2 \otimes \pi_3)$ of Weyl-type quality. Allowing $\pi_3$ to be an Eisenstein series we also obtain a Weyl-type subconvex bound for $L(1/2 + it, \pi_1 \otimes \pi_2)$.