Bounded equidistribution of special subvarieties II

TL;DR

This paper establishes lower bounds for Galois orbits of pure special subvarieties and introduces test invariants to prove bounded equidistribution for sequences of special subvarieties in mixed Shimura varieties, assuming GRH for CM fields.

Contribution

It extends equidistribution results to mixed Shimura varieties by defining test invariants and proving bounded equidistribution under GRH, generalizing previous pure cases.

Findings

Lower bounds for Galois orbits of pure special subvarieties.

Introduction of test invariants for non-pure special subvarieties.

Proof of bounded equidistribution under GRH for sequences with bounded test invariants.

Abstract

In this paper, we prove a lower bound for the Galois orbits of a pure special subvariety in a general mixed Shimura variety. For special subvarieties that are not pure, we propose the notion of test invariants as a substitute for the lower bound estimation, and prove the bounded equidistribution for sequences of special subvarieties with uniformly bounded test invariants in a given mixed Shimura variety. Both the estimation and the bounded equidistribution rely on the Generalized Riemann Hypothesis for CM fields, similar to the pure case treated by E. Ullmo and A. Yafaev as part of the ergodic-Galois alternative for the Andr\'e-Oort conjecture for pure Shimura varieties.