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 $L(u_j\times {\chi }_q,1/2){\ll }_{\epsilon }(q(1+|t_j|){)}^{1/3+\epsilon }$, 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 $|L(1/2+it,{\chi }_q)|{\ll }_{\epsilon }((1+|t|)q{)}^{1/6+\epsilon }$, 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.