# Subconvex equidistribution of cusp forms: reduction to Eisenstein observables

## TL;DR

The paper demonstrates that subconvexity for GL(1) twists implies subconvexity for PGL(2) twists of the adjoint L-function, linking quantum ergodicity to Eisenstein series testing.

## Contribution

It establishes a reduction from subconvexity problems for PGL(2) to those for GL(1), connecting different levels of automorphic forms and L-functions.

## Key findings

- H(GL(1)) implies H(PGL(2)) for subconvexity.
- Reduction links quantum ergodicity to Eisenstein series.
- Provides a new approach to subconvexity problems.

## 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 either of the groups GL(1) or 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.