CM-points and lattice counting on arithmetic compact Riemann surfaces
- Alsina Aubach, Montserrat
- Chatzakos, Dimitrios
Publication date
July 2020
DOI Publisher
Elsevier BV Journal
Journal of Number Theory Language
EnglishAbstract
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.Peer ReviewedPostprint (author's final draft
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