Counting orbits of integral points in families of  affinehomogeneous varieties and diagonal flows

Alexander Gorodnik; Frédéric Paulin

doi:10.3934/jmd.2014.8.25

Article Contents

2014, Volume 8, Issue 1: 25-59. Doi: 10.3934/jmd.2014.8.25

This issue Previous Article On Omri Sarig's work on the dynamics on surfaces Next Article Loci in strata of meromorphic quadratic differentials with fully degenerate Lyapunov spectrum

Counting orbits of integral points in families of affine homogeneous varieties and diagonal flows

Alexander Gorodnik^1, and
Frédéric Paulin^2,

1.
School of Mathematics, University of Bristol, Bristol BS8 1TW
2.
Département de mathématique, UMR 8628 CNRS, Bât. 425, Université Paris-Sud, 91405 ORSAY Cedex

Received: May 31, 2013

Published: June 2014

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Abstract

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 quadrics. Our arguments use the exponential mixing property of diagonal flows on homogeneous spaces.

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Mathematics Subject Classification: Primary: 37A17, 37A45; Secondary: 14M17, 20G20, 14G05, 11E20.

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