## Duke Mathematical Journal

### Subconvex equidistribution of cusp forms: Reduction to Eisenstein observables

Paul D. Nelson

#### Abstract

Let $\pi$ traverse a sequence of cuspidal automorphic representations of $\operatorname{GL}_{2}$ with large prime level, unramified central character, and bounded infinity type. For $G\in\{\operatorname{GL}_{1},\operatorname{PGL}_{2}\}$, 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 $H(\operatorname{GL}_{1})$ implies $H(\operatorname{PGL}_{2})$.

In geometric terms, $H(\operatorname{PGL}_{2})$ corresponds roughly to an instance of arithmetic quantum unique ergodicity with a power savings in the error term, $H(\operatorname{GL}_{1})$, 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
Revised: 10 January 2019
First available in Project Euclid: 12 June 2019

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

Digital Object Identifier
doi:10.1215/00127094-2019-0005

Mathematical Reviews number (MathSciNet)
MR3961213

#### 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

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