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2018 Height bounds and the Siegel property
Martin Orr
Algebra Number Theory 12(2): 455-478 (2018). DOI: 10.2140/ant.2018.12.455

Abstract

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 γG() 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.

Citation

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Martin Orr. "Height bounds and the Siegel property." Algebra Number Theory 12 (2) 455 - 478, 2018. https://doi.org/10.2140/ant.2018.12.455

Information

Received: 22 June 2017; Revised: 16 January 2018; Accepted: 15 February 2018; Published: 2018
First available in Project Euclid: 23 May 2018

zbMATH: 06880895
MathSciNet: MR3803710
Digital Object Identifier: 10.2140/ant.2018.12.455

Subjects:
Primary: 11F06
Secondary: 11G18

Rights: Copyright © 2018 Mathematical Sciences Publishers

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Vol.12 • No. 2 • 2018
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