Mass equidistribution for Poincar\'e series of large index

Let P₊, ₌ denote the Poincar\'e series of weight k and index m for the full modular group SL₂ (Z). Let \P₊, ₌\ be a sequence of Poincar\'e series for which m (k) satisfies m (k) / k and m (k) k^2 + 2{1 + 2 - } where 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 \P₊, ₌\ 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, P₊, ₌ vanishes exactly once at the cusp, which also implies that P₊, ₌ 0 in this range.