Subconvex equidistribution of cusp forms: Reduction to Eisenstein observables

Source abstract

Let π traverse a sequence of cuspidal automorphic representations of GL2 with large prime level, unramified central character, and bounded infinity type. For G∈{GL1,PGL2}, let H(G) denote the assertion that subconvexity holds for G-twists of the adjoint L-function of π, with polynomial dependence upon the conductor of the twist. We show that H(GL1) implies H(PGL2). In geometric terms, H(PGL2) corresponds roughly to an instance of arithmetic quantum unique ergodicity with a power savings in the error term, H(GL1), to the special case in which the relevant sequence of measures is tested against an Eisenstein series.