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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
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References:
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