Title: Mass equidistribution for Poincaré 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 $\mathrm{SL}_2(\mathbb{Z})$. Let $\{P_{k,m}\}$ be a sequence of Poincar\'e series for which $m(k)$ satisfies $m(k) / k \rightarrow\infty$ and $m(k) \ll k^{\frac{2 + 2\theta}{1 + 2\theta} - \epsilon}$ where $\theta$ is an exponent towards the Ramanujan Petersson conjecture. 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 metric 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 metric. Along the way we also improve a result of Rankin about the vanishing of Poincar\'e series. We show that for sufficiently large $k$ and $1\leq m\ll k^2$, $P_{k,m}$ vanishes exactly once at the cusp, which also implies that $P_{k,m}\not\equiv 0$ in this range.