Mass equidistribution for Poincar\'e series of large index

TL;DR

This paper proves that for a certain sequence of Poincaré series with large index, their mass and zeros become uniformly distributed on the modular surface as the weight increases, demonstrating a form of quantum unique ergodicity.

Contribution

It establishes mass equidistribution and zero distribution for Poincaré series with large index, extending understanding of their asymptotic behavior.

Findings

Mass of Poincaré series equidistributes on the modular surface.

Zeros of Poincaré series become uniformly distributed.

Results hold for sequences with index growing faster than the weight.

Abstract

Let $P_{k,m}$ denote the Poincar\'e series of weight $k$ and index $m$ for the full modular group $\mathrm{SL}_2(\mathbb{Z})$, and let $\{P_{k,m}\}$ be a sequence of Poincar\'e series for which $m(k)$ satisfies $m(k) / k \rightarrow\infty$ and $m(k) \ll k^{\frac{3}{2} - \epsilon}$. We prove that the $L^2$ mass of such a sequence equidistributes on $\mathrm{SL}_2(\mathbb{Z}) \backslash \mathbb{H}$ with respect to the hyperbolic measure as $k$ goes to infinity. As a consequence, we deduce that the zeros of such a sequence $\{P_{k,m}\}$ become uniformly distributed in $\mathrm{SL}_2(\mathbb{Z}) \backslash \mathbb{H}$ with respect to the hyperbolic measure.