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

Abstract

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 \times \chi_q, 1/2) \ll_{\varepsilon} (q (1 + |t_j|))^{1/3 + \varepsilon} , 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 |L(1/2+it, \chi_q)| \ll_{\varepsilon} ((1 + |t|) q)^{1/6 + \varepsilon} , 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.