The mixing conjecture of P. Michel and A. Venkatesh [in: Proceedings of the International Congress of Mathematicians (ICM), Madrid, Spain, 2006. Volume II: Invited lectures. Zürich: European Mathematical Society (EMS), 421–457 (2006; Zbl 1157.11019)] implies a special case of the following conjecture about equidistribution of Galois orbits of special points on products of modular curves: Let \(X\) be a finite product of complex modular curves. Let \(\{ x_i \}_{i}\) be a sequence of special points on \(X\), i.e., each coordinate of \(x_i\) is a CM point. Denote by \(\mu_i\) the normalized counting measure on the finite Galois orbit of \(x_i\). If the sequence \(\{ x_i \}\) has finite intersection with any proper special subvariety of X, then \(\{ \mu_i \}_{i}\) converges weak-\(*\) to the uniform probability measure on \(X\).
This conjecture implies the André-Oort conjecture for products of modular curves, which has been proved by J. Pila [Ann. Math. (2) 173, No. 3, 1779–1840 (2011; Zbl 1243.14022)]. The Andrée-Oort conjecture in this setting states that the sequence \(\{ x_i \}_i\) above must be Zariski dense in \(X\). In the present paper, the author studies the question of equidistribution of Galois orbits. He proves (conditionally) 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, he establishes (conditionally) for arbitrary products of indefinite Shimura curves the equidistribution of Galois orbits of generic sequences of CM points all of whose components have the same fundamental discriminant, assuming the CM fields are split at two fixed primes and have no exceptional zero.
In fact, Yu. V. Linnik [Ergodic properties of algebraic fields. Berlin: Springer-Verlag (1968; Zbl 0162.06801)] has proved Duke’s theorem about equidistribution of a sequence packets of CM points on the complex modular curve assuming that there is a fixed prime \(p\) that splits in all the CM fields in the sequence. In this proof Linnik used his ergodic method to bootstrap a weak bound on the self-correlation of the periodic measure on a toral packet in intermediate scales to full equidistribution using a dynamical argument. It is this dynamical argument where the assumption of a fixed split prime is used.
M. Einsiedler et al. [Ann. Math. (2) 173, No. 2, 815–885 (2011; Zbl 1248.37009)] have introduced a variant of Linnik’s ergodic method, which fits into the framework of homogeneous dynamics. The assumption of a fixed split prime \(p\) implies that the adelic periodic measures corresponding to the packets in the sequence are all invariant under a split \(p\)-adic torus. The approach of Einsiedler, Lindenstrauss, Michel and Venkatesh [loc. cit.] has significant ramifications when combined with the modern methods of measure rigidity for toral actions. Although measure rigidity requires further splitting assumptions, it can imply strong equidistribution results based on weaker arithmetic input compared to methods of harmonic analysis. This strategy is the starting point for the author’s proof. The assumption that two primes are split in all the CM fields in the sequence is required for the joinings theorem of Einsiedler and Lindenstrauss to apply. This measure rigidity result, however, falls short of demonstrating equidistribution due to the possibility of intermediate algebraic measures supported on Hecke correspondences and their translates. The author’s contribution is a method to exclude intermediate measures for toral periods. In particular, he uses 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.