## Duke Mathematical Journal

- Duke Math. J.
- Volume 168, Number 9 (2019), 1665-1722.

### 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.

#### Article information

**Source**

Duke Math. J., Volume 168, Number 9 (2019), 1665-1722.

**Dates**

Received: 14 February 2017

Revised: 10 January 2019

First available in Project Euclid: 12 June 2019

**Permanent link to this document**

https://projecteuclid.org/euclid.dmj/1560326499

**Digital Object Identifier**

doi:10.1215/00127094-2019-0005

**Mathematical Reviews number (MathSciNet)**

MR3961213

**Subjects**

Primary: 11F70: Representation-theoretic methods; automorphic representations over local and global fields

Secondary: 11F27: Theta series; Weil representation; theta correspondences 58J51: Relations between spectral theory and ergodic theory, e.g. quantum unique ergodicity

**Keywords**

subconvexity equidistribution quantum unique ergodicity L-functions automorphic forms Eisenstein series metaplectic group

#### Citation

Nelson, Paul D. Subconvex equidistribution of cusp forms: Reduction to Eisenstein observables. Duke Math. J. 168 (2019), no. 9, 1665--1722. doi:10.1215/00127094-2019-0005. https://projecteuclid.org/euclid.dmj/1560326499