Weyl-type hybrid subconvexity bounds for twisted $L$-functions and   Heegner points on shrinking sets
  We prove a Weyl-type subconvexity bound for the central value of the $L$-function of a Hecke-Maass form or a holomorphic Hecke eigenform twisted by a quadratic Dirichlet character, uniform in the archimedean parameter as well as the twisting parameter. A similar hybrid bound holds for quadratic Dirichlet $L$-functions, improving on a result of Heath-Brown.   As a consequence of these new bounds, we obtain explicit estimates for the number of Heegner points of large odd discriminant in shrinking sets. 
Matthew P. Young
math.NT