Mass equidistribution for Poincaré series of large index

Abstract

Let $P_{k,m}$ denote the Poincaré 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 Poincaré series for which $m(k)$ satisfies $m(k)/k\to \infty$ 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.