Weyl-type Hybrid Subconvexity Bounds for Twisted L-functions and Heegner Points on Shrinking Sets

Let q be odd and squarefree, and let χq be the quadratic Dirichlet character of conductor q. Let uj be a Hecke-Maass cusp form on Γ0(q) with spectral parameter tj . By an extension of work of Conrey and Iwaniec, we show L(uj ×χq, 1/2) ≪ε (q(1 + |tj |))1/3+ε, uniformly in both q and tj . A similar bound holds for twists of a holomorphic Hecke cusp form of large weight k. Furthermore, we show that |L(1/2 + it, χq)| ≪ε ((1 + |t|)q)1/6+ε, 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.