The hyperbolic circle problem over Heegner points

TL;DR

This paper improves the error bounds in counting Heegner points within hyperbolic circles for the full modular group, using advanced number theoretic techniques and new fractional moment estimates.

Contribution

It provides a logarithmic improvement on Selberg's bound for the hyperbolic circle problem over Heegner points, introducing new fractional moment estimates and leveraging Waldspurger's formula.

Findings

Logarithmic improvement on error bounds

Enhanced understanding of Heegner point distribution

Novel fractional moment estimate

Abstract

For the full modular group, we obtain a logarithmic improvement on Selberg's long-standing bound for the error term of the counting function in the hyperbolic circle problem over Heegner points of different discriminants. The main ingredients in our method are Waldspurger's formula, twisted first moments of certain Rankin-Selberg convolutions, and a new fractional moment estimate.