Subconvex equidistribution of cusp forms: Reduction to Eisenstein observables

Abstract

Let $\pi$ traverse a sequence of cuspidal automorphic representations of ${GL}_2$ with large prime level, unramified central character, and bounded infinity type. For $G\in \{{GL}_1,{PGL}_2\}$, let $H(G)$ denote the assertion that subconvexity holds for $G$-twists of the adjoint $L$-function of $\pi$, with polynomial dependence upon the conductor of the twist. We show that $H({GL}_1)$ implies $H({PGL}_2)$.

In geometric terms, $H({PGL}_2)$ corresponds roughly to an instance of arithmetic quantum unique ergodicity with a power savings in the error term, $H({GL}_1)$, to the special case in which the relevant sequence of measures is tested against an Eisenstein series.