The hyperbolic circle problem

Let $H$ be the hyperbolic plane and $, ∈ H$ two points. For $Γ = SL2(Z)$ the hyperbolic circle problem asks to estimate the number of translates of $w$ by elements of $Γ$ lie in a hyperbolic ball centered in $z$ of volume $X$, in the limit in which the volume tends to infinity. Geometric tiling arguments fail due to the hyperbolic geometry of the space. Thus, spectral arguments are required to obtain an estimate. We present a recent result of Chatzakos, Cherubini, Lester, Risager, in which they improve the original bound of Selberg in the case of $z,w$ Heegner points of different discriminant. The work uses Waldspurger formula to relate spectral sums bounds to nontrivial estimates of fractional moments of Rankin-Selberg L-functions. ​

Let $H$ be the hyperbolic plane and $, ∈ H$ two points. For $Γ = SL2(Z)$ the hyperbolic circle problem asks to estimate the number of translates of $w$ by elements of $Γ$ lie in a hyperbolic ball centered $z$ in of volume $X$, in the limit in which the volume tends to infinity. Geometric tiling arguments fail due to the hyperbolic geometry of the space. Thus, spectral arguments are required to obtain an estimate. We present a recent result of Chatzakos, Cherubini, Lester, Risager, in which they improve the original bound of Selberg in the case of $z,w$ Heegner points of different discriminant. The work uses Waldspurger formula to relate spectral sums bounds to nontrivial estimates of fractional moments of Rankin-Selberg L-functions. ​