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Height bounds and the Siegel property
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Orr, Martin (2018) Height bounds and the Siegel property. Algebra & Number Theory, 12 (2). pp. 455-478. doi:10.2140/ant.2018.12.455 ISSN 1937-0652.
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Official URL: http://dx.doi.org/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 ∈ 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 .
Item Type: | Journal Article | |||||||||
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Subjects: | Q Science > QA Mathematics | |||||||||
Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | |||||||||
Library of Congress Subject Headings (LCSH): | Siegel domains, Polynomials | |||||||||
Journal or Publication Title: | Algebra & Number Theory | |||||||||
Publisher: | Mathematical Sciences Publishers | |||||||||
ISSN: | 1937-0652 | |||||||||
Official Date: | 13 May 2018 | |||||||||
Dates: |
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Volume: | 12 | |||||||||
Number: | 2 | |||||||||
Page Range: | pp. 455-478 | |||||||||
DOI: | 10.2140/ant.2018.12.455 | |||||||||
Status: | Peer Reviewed | |||||||||
Publication Status: | Published | |||||||||
Access rights to Published version: | Restricted or Subscription Access | |||||||||
Copyright Holders: | Mathematical Sciences Publishers | |||||||||
Date of first compliant deposit: | 31 July 2018 | |||||||||
Date of first compliant Open Access: | 6 August 2018 | |||||||||
RIOXX Funder/Project Grant: |
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