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.