Weyl-type hybrid subconvexity bounds for twisted L-functions and Heegner points on shrinking sets

Let q be odd and squarefree, and let chi(q) be the quadratic Dirichlet character of conductor q. Let u(j) be a Hecke-Maass cusp form on Gamma(0) (q) with spectral parameter t(j). By an extension of work of Conrey and Iwaniec, we show L(u(j) x chi(q) ,1/2) <<(epsilon) q(1 + vertical bar t(j)vertical bar))(1/3+epsilon) , uniformly in both q and t(j). A similar bound holds for twists of a holomorphic Hecke cusp form of large weight k. Furthermore, we show that vertical bar L(1/2 + it, chi(q))vertical bar <<(epsilon) ((1 + vertical bar t vertical bar)(q))(1/6+epsilon) , 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.