Height bounds and the Siegel property

Orr, Martin

doi:10.2140/ant.2018.12.455

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Height bounds and the Siegel property

Martin Orr

Vol. 12 (2018), No. 2, 455–478

DOI: 10.2140/ant.2018.12.455

Abstract

Let $G$ be a reductive group defined over $ℚ$ and let $S$ be a Siegel set in $G (ℝ)$ . The Siegel property tells us that there are only finitely many $γ \in G (ℚ)$ of bounded determinant and denominator for which the translate $γ . S$ intersects $S$ . We prove a bound for the height of these $γ$ 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 $G (ℚ)$ -translates of a Siegel set for $G$ .

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Keywords

reduction theory, Siegel sets, unlikely intersections

Mathematical Subject Classification 2010

Primary: 11F06

Secondary: 11G18

Milestones

Received: 22 June 2017

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

Vol. 12, No. 2, 2018