Averaging over Heegner points in the hyperbolic circle problem

For $Gamma={hbox{PSL}_2( {mathbb Z})}$ 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 $D$, Selbergs 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 Lindelof conjecture for twists of the $L$-functions attached to Maa{ss} cusp forms.