The mixing conjecture under GRH

TL;DR

This work proves the Mixing Conjecture of Michel–Venkatesh for the class-group action on Linnik-type Heegner data, conditional on GRH and, in the indefinite quaternion case, on Ramanujan bounds for PGL$_2$. The authors develop a comprehensive adelic framework for Heegner packets, joint toral data, and level structures, and they derive effective quantitative rates by blending Weyl’s equidistribution with spectral theory, Waldspurger-type formulae, and Rankin–Selberg/L-function bounds. Two complementary regimes—large $q$ and small $q$—are analyzed via different strategies: a large-$q$ diagonal/ off-diagonal approach using automorphic $L$-functions, and a small-$q$ analysis via spectral decomposition across intermediate Hecke correspondences; both are tied together through strong Gelfand formations. The results yield equidistribution statements for joint Heegner data in quaternionic Shimura-type varieties and connect to André–Oort-type phenomena, with explicit, effective convergence rates depending on the discriminant and the representative norm $q$.

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.

Theorems & Definitions (92)

• Conjecture 1: Michel--Venkatesh
• Theorem 1.1: Main Theorem, first version
• Theorem 2.1: Main theorem, second version
• Remark 2.1
• Example 1
• Theorem 2.2: Main Theorem: third version
• Remark 2.2
• Remark 2.3
• Remark 3.1
• Lemma 3.1