CM-points and lattice counting on arithmetic compact Riemann surfaces

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Let X(D,1)=Γ(D,1)\\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→−∞. We prove that if |d| is sufficiently large compared to the radius r≈log⁡X of the circle, we can improve on the classical O(X2/3)-bound of Selberg. Our result extends the result of Petridis and Risager for the modular surface to arithmetic compact Riemann surfaces.