Vol. 12, No. 2, 2018

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Height bounds and the Siegel property

Martin Orr

Vol. 12 (2018), No. 2, 455–478

Let G be a reductive group defined over and let S be a Siegel set in G(). The Siegel property tells us that there are only finitely many γ G() 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 GL2, 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()-translates of a Siegel set for G.

reduction theory, Siegel sets, unlikely intersections
Mathematical Subject Classification 2010
Primary: 11F06
Secondary: 11G18
Received: 22 June 2017
Revised: 16 January 2018
Accepted: 15 February 2018
Published: 13 May 2018
Correction: 26 May 2023
Martin Orr
Department of Mathematics
Imperial College London
South Kensington
United Kingdom