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Height bounds and the Siegel property. (English) Zbl 1442.11074
Summary: Let $$\mathbf G$$ be a reductive group defined over $$\mathbb Q$$ and let $$\mathfrak S$$ be a Siegel set in $$\mathbf G(\mathbb R)$$. The Siegel property tells us that there are only finitely many $$\gamma\in\mathbf G(\mathbb Q)$$ of bounded determinant and denominator for which the translate $$\gamma.\mathfrak S$$ intersects $$\mathfrak S$$. We prove a bound for the height of these $$\gamma$$ which is polynomial with respect to the determinant and denominator. The bound generalises a result of Habegger and Pila dealing with the case of $$\mathrm{GL}_2$$, and has applications to the Zilber-Pink conjecture on unlikely intersections in Shimura varieties.
In addition we prove that if $$\mathbf H$$ is a subset of $$\mathbf G$$, then every Siegel set for $$\mathbf H$$ is contained in a finite union of $$\mathbf G(\mathbb Q)$$-translates of a Siegel set for $$\mathbf G$$.

##### MSC:
 11F06 Structure of modular groups and generalizations; arithmetic groups 11G18 Arithmetic aspects of modular and Shimura varieties
##### Keywords:
reduction theory; Siegel sets; unlikely intersections
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##### References:
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