Averaging over Heegner points in the hyperbolic circle problem

Abstract

For $\Gamma ={PSL}_2(\mathbb{Z})$ the hyperbolic circle problem aims to estimate the number of elements of the orbit $\Gamma z$ inside the hyperbolic disc centered at $z$ with radius ${\cosh }^{-1}(X/2)$. We show that, by averaging over Heegner points $z$ of discriminant $D$, Selberg's error term estimate can be improved, if $D$ is large enough. The proof uses bounds on spectral exponential sums, and results towards the sup-norm conjecture of eigenfunctions, and the Lindelöf conjecture for twists of the $L$-functions attached to Maaß cusp forms.

20 pages. Fixed a few typos and minor inaccuracies