Mass equidistribution for Poincar\'e series of large index

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.