Height bounds and the Siegel property

TL;DR

This paper establishes polynomial height bounds for certain rational points related to Siegel sets in reductive groups, extending previous results and applying to conjectures in arithmetic geometry.

Contribution

It generalizes height bounds for Siegel set translates from $ ext{GL}_2$ to general reductive groups and explores their implications for the Zilber--Pink conjecture.

Findings

Proved polynomial height bounds for $ ext{G}(Q)$ elements intersecting Siegel sets.

Extended Habegger and Pila's results from $ ext{GL}_2$ to arbitrary reductive groups.

Showed Siegel sets for subgroups are contained in finitely many translates of larger Siegel sets.

Abstract

Let $\mathbf{G}$ be a reductive group defined over $\mathbb{Q}$ and let $\mathfrak{S}$ be a Siegel set in $\mathbf{G}(\mathbb{R})$. The Siegel property tells us that there are only finitely many $\gamma \in \mathbf{G}(\mathbb{Q})$ of bounded determinant and denominator for which the translate $\gamma.\mathfrak{S}$ intersects $\mathfrak{S}$. We prove a bound for the height of these $\gamma$ which is polynomial with respect to the determinant and denominator. The bound generalises a result of Habegger and Pila dealing with the case of $\mathbf{GL}_2$, and has applications to the Zilber--Pink conjecture on unlikely intersections in Shimura varieties. In addition we prove that if $\mathbf{H}$ is a subgroup of $\mathbf{G}$, then every Siegel set for $\mathbf{H}$ is contained in a finite union of $\mathbf{G}(\mathbb{Q})$-translates of a Siegel set for $\mathbf{G}$).