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

Alexander Gorodnik (1) , Frédéric Paulin (2)

1 School of Mathematics [Bristol]
2 LM-Orsay - Laboratoire de Mathématiques d'Orsay

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.

Mots clés: integral point; homogeneous variety; Siegel weight; counting; decomposable form; norm form; diagonalisable flow; mixing; exponential decay of correlation

Domaines: Systèmes dynamiques [math.DS]; Théorie des groupes [math.GR]; Théorie des nombres [math.NT]

Alexander Gorodnik, Frédéric Paulin. Counting orbits of integral points in families of affine homogeneous varieties and diagonal flows. 2011. ⟨hal-00635948v2⟩