CM-points and Lattice counting on arithmetic compact Riemann surfaces

TL;DR

This paper investigates the distribution of Heegner points on arithmetic compact Riemann surfaces and improves classical bounds on the error term in the hyperbolic circle problem for large discriminants.

Contribution

It extends the improved error bounds for the hyperbolic circle problem from modular surfaces to more general arithmetic compact Riemann surfaces.

Findings

Improved error bounds for large discriminants compared to classical results.

Extension of Petridis and Risager's results to arithmetic compact Riemann surfaces.

Analysis of Heegner points distribution on Shimura curves.

Abstract

Let $X(D,1) =\Gamma(D,1) \backslash \mathbb{H}$ denote the Shimura curve of level $N=1$ arising from an indefinite quaternion algebra of fixed discriminant $D$. We study the discrete average of the error term in the hyperbolic circle problem over Heegner points of discriminant $d <0$ on $X(D,1)$ as $d \to -\infty$. We prove that if $|d|$ is sufficiently large compared to the radius $r \approx \log X$ of the circle, we can improve on the classical $O(X^{2/3})$-bound of Selberg. Our result extends the result of Petridis and Risager for the modular surface to arithmetic compact Riemann surfaces.