## Algebra & Number Theory

### Height bounds and the Siegel property

Martin Orr

#### 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 $GL2$, 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 $G(ℚ)$-translates of a Siegel set for $G$.

#### Article information

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

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

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

#### 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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