#### Vol. 12, No. 2, 2018

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Height bounds and the Siegel property

### Martin Orr

Vol. 12 (2018), No. 2, 455–478
##### Abstract

Let $G$ be a reductive group defined over $ℚ$ and let $\mathfrak{S}$ be a Siegel set in $G\left(ℝ\right)$. The Siegel property tells us that there are only finitely many $\gamma \in G\left(ℚ\right)$ 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 ${GL}_{2}$, and has applications to the Zilber–Pink conjecture on unlikely intersections in Shimura varieties.

In addition we prove that if $H$ is a subset of $G$, then every Siegel set for $H$ is contained in a finite union of $G\left(ℚ\right)$-translates of a Siegel set for $G$.

##### Keywords
reduction theory, Siegel sets, unlikely intersections
Primary: 11F06
Secondary: 11G18
##### Milestones
Revised: 16 January 2018
Accepted: 15 February 2018
Published: 13 May 2018
##### Authors
 Martin Orr Department of Mathematics Imperial College London South Kensington London United Kingdom