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 .
"Height bounds and the Siegel property." Algebra Number Theory 12 (2) 455 - 478, 2018. https://doi.org/10.2140/ant.2018.12.455