Algebra & Number Theory
- Algebra Number Theory
- Volume 12, Number 2 (2018), 455-478.
Height bounds and the Siegel property
Let be a reductive group defined over and let be a Siegel set in . The Siegel property tells us that there are only finitely many of bounded determinant and denominator for which the translate intersects . We prove a bound for the height of these which is polynomial with respect to the determinant and denominator. The bound generalises a result of Habegger and Pila dealing with the case of , and has applications to the Zilber–Pink conjecture on unlikely intersections in Shimura varieties.
In addition we prove that if is a subset of , then every Siegel set for is contained in a finite union of -translates of a Siegel set for .
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
Permanent link to this document
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]
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