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.
Highlights
We investigate the asymptotic distribution of integral points on families of homogeneous algebraic varieties using dynamical systems techniques
The aim of this paper is to develop a direct argument that establishes an asymptotic formula for the quantities β(v) on average
We show that sums over β(v) can be interpreted as volumes of intersections of two transversal submanifolds in a suitable homogeneous space
Summary
We investigate the asymptotic distribution of integral points on families of homogeneous algebraic varieties using dynamical systems techniques. There exist c = c(q, l) > 0 and δ = δ(q) > 0 such that, as r → +∞, vol (Lu(Z) ∩ L(R)0)\(Lu ∩ L(R)0) = c rn−2 + O rn−2−δ This result fits into the above program (up to a slight modification of the Siegel weights, see Section 2), by taking V = Cn, Λ = Zn, and π : L → GL(V) the inclusion map, noting that L is semisimple, and defined and anisotropic over Q as a consequence of the assumptions (see Section 3.2 for details, where we explicit c). We note that we do not assume Xv(R) to be an affine symmetric space or that the stabiliser is a maximal subgroup, contrarily to [DRS] and many other references Another difference with the counting results of [EMS, Oh1, EO] is that these papers are using the dynamics of unipotent flows, as instead we are using here the mixing property with exponential decay of correlations of diagonalisable flows, in the spirit of [KM1] (see [EM, BO]). We are using the proof of the main result of [PP] as a guideline
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