Averaging over Heegner points in the hyperbolic circle problem

For $Γ={\hbox{PSL}_2( {\mathbb Z})}$ the hyperbolic circle problem aims to estimate the number of elements of the orbit $Γz$ 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.

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