Mass equidistribution for Poincar\'e series of large index

Let Pk,m denote the Poincar\'e series of weight k and index m for the full modular group SL₂(Z). Let ,m\ be a sequence of Poincar\'e series for which $m(k)$ satisfies m(k) / k →∞ and m(k) k2 + 2θ/1 + 2θ - ε where θ is an exponent towards the Ramanujan Petersson conjecture. We prove that the L² mass of such a sequence equidistributes on SL₂(Z) H with respect to the hyperbolic metric as k goes to infinity. As a consequence, we deduce that the zeros of such a sequence ,m\ become uniformly distributed in SL₂(Z) 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≤ m k², Pk,m vanishes exactly once at the cusp, which also implies that Pk,m≡ 0 in this range.