Height bounds and the Siegel property

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 $\mathbf{G}$).