Algebra & Number Theory

Height bounds and the Siegel property

Martin Orr

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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.

Article information

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

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

Permanent link to this document
https://projecteuclid.org/euclid.ant/1527040851

Digital Object Identifier
doi:10.2140/ant.2018.12.455

Mathematical Reviews number (MathSciNet)
MR3803710

Zentralblatt MATH identifier
06880895

Subjects
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]

Keywords
reduction theory Siegel sets unlikely intersections

Citation

Orr, Martin. Height bounds and the Siegel property. Algebra Number Theory 12 (2018), no. 2, 455--478. doi:10.2140/ant.2018.12.455. https://projecteuclid.org/euclid.ant/1527040851


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