Journal of Modern Dynamics (JMD)

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

Pages: 25 - 59, Issue 1, March 2014      doi:10.3934/jmd.2014.8.25

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Alexander Gorodnik - School of Mathematics, University of Bristol, Bristol BS8 1TW, United Kingdom (email)
Frédéric Paulin - Département de mathématique, UMR 8628 CNRS, Bât. 425, Université Paris-Sud, 91405 ORSAY Cedex, France (email)

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.

Keywords:  Integral point, homogeneous variety, Siegel weight, counting, decomposable form, norm form, diagonalizable flow, mixing, exponential decay of correlation.
Mathematics Subject Classification:  Primary: 37A17, 37A45; Secondary: 14M17, 20G20, 14G05, 11E20.

Received: June 2013;      Available Online: July 2014.