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

Abstract

Let S be a Shimura variety. We conjecture that the heights of special points in \(S(\overline{\mathbb {Q}})\) are discriminant negligible with respect to some Weil height function \(h:S(\overline{\mathbb {Q}})\rightarrow \mathbb {R}\). 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 particular, we give a new proof of a theorem of Tsimerman on lower bounds for Galois degrees of special points in Shimura varieties of abelian type. This gives a new proof of the André–Oort conjecture for such varieties that avoids the use of Masser–Wüstholz isogeny estimates, replacing them by a point-counting argument.