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( √ −D). For f a weight zero Hecke-Maass newform of level q and Θχ the weight one theta series of level D corresponding to an ideal class group character χ of Q( √ −D), we establish a hybrid subconvexity bound for L(f×Θχ, s) at s = 1/2 when q D for 0 < η < 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 →∞ for q ≤ D1/20−ε. Our approach to these problems is connected to estimating the L-restriction norm of a Maass form of large level q 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.