Mathematics – Dynamical Systems
Scientific paper
2011-10-26
Mathematics
Dynamical Systems
33 pages
Scientific paper
In this paper, we study the distribution of integral points on parametric families of affine homogeneous varieties. By the work of Borel and Harish-Chandra, the set of integral points on each such variety consists of finitely many orbits of arithmetic groups, and we establish an asymptotic formula (on average) for the number of the orbits indexed by their Siegel weights. In particular, we deduce asymptotic formulas for the number of inequivalent integral representations by decomposable forms and by norm forms in division algebras, and for the weighted number of equivalence classes of integral points on sections of quadratic surfaces. Our arguments use the exponential mixing property of diagonal flows on homogeneous spaces.
Gorodnik Alexander
Paulin Frédéric
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