Subconvexity and equidistribution of Heegner points in the level aspect

Let q be a prime and -D < -4 be an odd fundamental discriminant such that q splits in Q(\sqrt{-D}). For f a weight zero Hecke-Maass newform of level q and h the weight one theta series of level D corresponding to an ideal class group character of Q(\sqrt{-D}), we establish a hybrid subconvexity bound for L(f \times h,s) at the central point when q = D^{\eta} for 0 < \eta < 1. With this circle of ideas, we show that the Heegner points of level q and discriminant D become equidistributed, in a natural sense, as q, D become large with q < D^{1/20-\varepsilon}. Our approach to these problems is connected to estimating the L^2-restriction norm of a Maass form of large level when restricted to the collection of Heegner points. We furthermore establish bounds for quadratic twists of Hecke-Maass L-functions with simultaneously large level and large quadratic twist, and hybrid bounds for quadratic Dirichlet L-functions in certain ranges.