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## CM-points and lattice counting on arithmetic compact Riemann surfaces

Author
Alsina, M.; Chatzakos, D.
Type of activity
Journal article
Journal
Journal of number theory
Date of publication
2020-07
Volume
212
First page
339
Last page
353
DOI
10.1016/j.jnt.2019.11.009
Project funding
The conjecture of Birch and Swinnerton-Dyer
Repository
http://hdl.handle.net/2117/179201
https://arxiv.org/pdf/1808.01318.pdf
URL
https://www.sciencedirect.com/science/article/pii/S0022314X19304093
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...
Citation
Alsina, M.; Chatzakos, D. CM-points and lattice counting on arithmetic compact Riemann surfaces. "Journal of number theory", July 2020, vol. 212, p. 339-353.
Keywords
Arithmetic groups, Discontinuous groups and automorphic forms, Spectral theory
Group of research
TN - Number Theory Research Group