Lower bounds for Galois orbits of special points on Shimura varieties: a point-counting approach

Abstract

Let $S$ be a Shimura variety and let $h$ be a Weil height function on $S$. We conjecture that the heights of special points in $S$ are discriminant negligible. Assuming this conjecture to be true, we prove that the sizes of the Galois orbits of special points grow as a fixed power of their discriminant (an invariant we will define in the text). In the case of Shimura varieties of abelian type, the height bound holds by the recently proved averaged Colmez formula, and our theorem gives a new proof of Tsimerman's Galois lower bound in this case. The main novelty is that our approach avoids the use of Masser-W\"ustholz isogeny estimates, replacing them by a point-counting argument, and establishes lower bounds for Galois orbits conditional on height bounds for \emph{arbitrary} Shimura varieties. In particular, following the Pila-Zannier strategy (and Gao's work in the mixed case) this implies that the Andre-Oort conjecture for an arbitrary (mixed) Shimura variety follows from the corresponding conjecture on heights of special points.