The mixing conjecture under GRH

TL;DR

This paper proves the Mixing Conjecture for class group actions on Heegner points on arithmetic surfaces, conditional on GRH and Ramanujan conjecture, using spectral theory and analytic number theory techniques.

Contribution

It provides a conditional proof of the Mixing Conjecture with an effective rate, advancing understanding of automorphic forms and class group actions.

Findings

Conditional proof of the Mixing Conjecture under GRH

Effective rate of mixing established

Application of spectral theory and L-functions techniques

Abstract

We prove the Mixing Conjecture of Michel--Venkatesh for the class group action on Heegner points of large discriminant on compact arithmetic surfaces attached to maximal orders in rational quaternion algebras. The proof is conditional on the Generalized Riemann Hypothesis, and when the division algebra is indefinite we furthermore assume the Ramanujan conjecture. Our methods, which provide an effective rate, are based on the spectral theory of automorphic forms and their $L$-functions, as well as techniques in classical analytic number theory.