Quantitative equidistribution on hyperbolic surfaces and arithmetic applications

Abstract

The Wasserstein distance quantifies the distance between two probability measures on a metric space. We prove an analogue of the Berry–Esseen inequality for the Wasserstein distance on a finite area hyperbolic surface. This inequality controls the Wasserstein distance via an average of Weyl sums, which are integrals of Maaß cusp forms and Eisenstein series with respect to these probability measures. As applications, we prove upper bounds for the Wasserstein distance for some equidistribution problems on the modular surface ${\mathrm{SL}}_2(\mathbb{Z})\backslash \mathbb{H}$, namely Duke’s theorems on the equidistribution of Heegner points and of closed geodesics and Watson’s theorem on the mass equidistribution of Hecke–Maaß cusp forms conditionally under the assumption of the generalised Lindelöf hypothesis.