Joint Equidistribution of CM Points

TL;DR

This paper proves the mixing conjecture for CM points on Shimura curves under certain conditions, establishing their equidistribution and introducing a new method to exclude intermediate measures using geometric and analytic techniques.

Contribution

It introduces a novel method to exclude intermediate measures for toral periods by geometric expansion of cross-correlation, advancing understanding of equidistribution of CM points.

Findings

Proves the mixing conjecture for toral packets with negative fundamental discriminants.

Establishes equidistribution of Galois orbits of CM points on Shimura curves.

Develops a new technique to bound shifted convolution sums for polynomials in two variables.

Abstract

We prove the mixing conjecture of Michel and Venkatesh for toral packets with negative fundamental discriminants and split at two fixed primes; assuming all splitting fields have no exceptional Landau-Siegel zero. As a consequence we establish for arbitrary products of indefinite Shimura curves the equidistribution of Galois orbits of generic sequences of CM points all whose components have the same fundamental discriminant; assuming the CM fields are split at two fixed primes and have no exceptional zero. The joinings theorem of Einsiedler and Lindenstrauss applies to the toral orbits arising in these results. Yet it falls short of demonstrating equidistribution due to the possibility of intermediate algebraic measures supported on Hecke correspondences and their translates. The main novel contribution is a method to exclude intermediate measures for toral periods. The crux is a geometric expansion of the cross-correlation between the periodic measure on a torus orbit and a Hecke correspondence, expressing it as a short shifted convolution sum. The latter is bounded from above generalizing the method of Shiu and Nair to polynomials in two variables on smooth domains.