[Paper Review] Counting orbits of integral points in families of affine homogeneous varieties and diagonal flows

Alexander Gorodnik, Frédéric Paulin|Bristol Research (University of Bristol)|Oct 26, 2011

Advanced Algebra and Geometry61 references6 citations

TL;DR

This paper establishes an asymptotic formula for the number of orbits of integral points on families of affine homogeneous varieties, weighted by Siegel weights, using exponential mixing of diagonal flows on homogeneous spaces. The key result shows that the sum of weighted orbits grows exponentially with a precise error term, generalizing classical counting results in number theory and dynamics.

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

Motivation & Objective

• To understand the asymptotic distribution of integral points on parametric families of affine homogeneous varieties.
• To address the limitation of classical methods like the Hardy–Littlewood circle method, which do not apply in general cases.
• To extend counting results beyond semi-simple groups to include non-semi-simple cases, broadening applicability.
• To provide an asymptotic formula for weighted orbit counts using dynamical systems techniques.
• To establish a general framework for counting orbits with Siegel weights in families of varieties via equidistribution of submanifolds.

Proposed method

• Interpret sums over weighted orbits as volumes of intersections of transversal submanifolds in a homogeneous space.
• Use the exponential mixing property of diagonal flows on homogeneous spaces to deduce equidistribution of one submanifold.
• Apply the relative Langlands decomposition of parabolic subgroups to parametrize the family of varieties.
• Define Siegel weights via volume ratios of arithmetic quotients of stabilizer subgroups and the full group.
• Use a Riemannian metric on the real points of the group to normalize measures and ensure orthogonality of Lie algebras.
• Establish asymptotic equivalence by relating the counting sum to the exponential growth of the diagonal flow parameter $ s $, with error term $ O(e^{( heta - heta_0)t}) $.

Experimental results

Research questions

• RQ1 How can one uniformly count orbits of integral points on families of affine homogeneous varieties when classical methods fail?
• RQ2 What is the asymptotic behavior of the weighted sum of orbits, where weights are given by Siegel weights?
• RQ3 Can the exponential mixing of diagonal flows be used to derive asymptotic formulas for orbit counting in non-semi-simple settings?
• RQ4 How does the equidistribution of submanifolds under diagonal flows lead to precise counting estimates?
• RQ5 What is the precise growth rate of the number of primitive integral points on orbits parametrized by a one-parameter diagonal flow?

Key findings

• The sum of weighted orbits over $ 0 o s o t $ grows asymptotically as $ c e^{ heta t} + O(e^{( heta - heta_0)t}) $, where $ heta = ext{tr}( ext{Ad}(a_1)|_{rak{u}}) $, with $ c, heta_0 > 0 $.
• The main result applies to reductive groups $ f L $ defined and anisotropic over $ bQ $, with a rational representation $ ho $ and a one-parameter diagonal flow $ (a_s) $.
• For the special case of integral points in orders of central simple algebras, the number of $ bO^ imes $-orbits of elements with norm in $ [1, r] $ is asymptotically $ c heta(n^2) r^n (1 + O(r^{- heta_0})) $.
• The Siegel weights are constant on orbits when the stabilizer is trivial, simplifying the counting in symmetric cases.
• The proof establishes that $ f LP $ is Zariski-open in $ f G $, ensuring the family of varieties is well-parametrized.
• The error term $ O(e^{( heta - heta_0)t}) $ is uniform and quantitatively controlled via exponential mixing of the flow.