CM-points and lattice counting on arithmetic compact Riemann surfaces

descriptionPublicationkeyboard_double_arrow_right Article , Preprint 01 Jul 2020Embargo end date: 01 Jan 2018 English Publisher:Elsevier BVJournal:Journal of Number Theory, volume 212, pages 339-353 (issn: 0022-314X,

Copyright policy )Funded by:EC | NODAL

Authors: Alsina Aubach, Montserrat; Chatzakos, Dimitrios;

doi: 10.1016/j.jnt.2019.11.009 , 10.48550/arxiv.1808.01318

arXiv: 1808.01318

handle: 2117/179201

CM-points and lattice counting on arithmetic compact Riemann surfaces

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Abstract

Let $X(D,1) =��(D,1) \backslash \mathbb{H}$ 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.

10 pages; Final version to appear in Journal of Number Theory

Related Organizations

Universitat Politècnica de Catalunya
Spain
Universitat Polite`cnica de Catalunya
Spain
UNIVERSITE LILLE 1
France
King's College London
United Kingdom

Keywords

Orbit growth in dynamical systems, Automorphisms, Spectral theory; trace formulas (e.g., that of Selberg), Automorfismes, Superfícies de, Arithmetic groups, 11F72 (primary), 37C35, 37D40 (secondary), :37 Dynamical systems and ergodic theory::37C Smooth dynamical systems: general theory [Classificació AMS], Classificació AMS::37 Dynamical systems and ergodic theory::37D Dynamical systems with hyperbolic behavior, FOS: Mathematics, Riemann, Number Theory (math.NT), Àrees temàtiques de la UPC::Matemàtiques i estadística, Mathematics - Number Theory, Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.), discontinuous groups and automorphic forms, :Matemàtiques i estadística [Àrees temàtiques de la UPC], spectral theory, arithmetic groups, Classificació AMS::37 Dynamical systems and ergodic theory::37C Smooth dynamical systems: general theory, Teoria espectral (Matemàtica), Riemann surfaces, :37 Dynamical systems and ergodic theory::37D Dynamical systems with hyperbolic behavior [Classificació AMS], Discontinuous groups and automorphic forms, Spectral theory (Mathematics), Riemann, Superfícies de, Spectral theory

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