Algebra & Number Theory

Height bounds and the Siegel property

Martin Orr

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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.

Article information

Algebra Number Theory, Volume 12, Number 2 (2018), 455-478.

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

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11F06: Structure of modular groups and generalizations; arithmetic groups [See also 20H05, 20H10, 22E40]
Secondary: 11G18: Arithmetic aspects of modular and Shimura varieties [See also 14G35]

reduction theory Siegel sets unlikely intersections


Orr, Martin. Height bounds and the Siegel property. Algebra Number Theory 12 (2018), no. 2, 455--478. doi:10.2140/ant.2018.12.455.

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