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

TL;DR

This paper establishes new Weyl-type subconvexity bounds for twisted $L$-functions, leading to improved estimates for the distribution of Heegner points in shrinking regions.

Contribution

It introduces a novel hybrid subconvexity bound for twisted $L$-functions that is uniform in key parameters, advancing understanding of their behavior.

Findings

Weyl-type subconvexity bounds for twisted $L$-functions

Improved estimates for Heegner points in shrinking sets

Enhanced bounds for quadratic Dirichlet $L$-functions

Abstract

We prove a Weyl-type subconvexity bound for the central value of the $L$-function of a Hecke-Maass form or a holomorphic Hecke eigenform twisted by a quadratic Dirichlet character, uniform in the archimedean parameter as well as the twisting parameter. A similar hybrid bound holds for quadratic Dirichlet $L$-functions, 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.