[Paper Review] Manin's conjecture on rational points of bounded height and adelic mixing

Alex Gorodnik, François Maucourant+1 more|arXiv (Cornell University)|Jan 6, 2006

Advanced Algebra and Geometry43 references12 citations

TL;DR

This paper proves Manin's conjecture for the wonderful compactification of a connected adjoint semisimple algebraic group G over a number field K by establishing the asymptotic count of K-rational points of bounded height in any irreducible representation. Using an L²-mixing argument with a quantitative rate, the authors further compute the explicit asymptotic distribution of rational points on the adelic space X(A), confirming Peyre's prediction.

ABSTRACT

Dedicated to Prof. Gregory Margulis on the occasion of his sixtieth birthday Abstract. Let K be a number field. We compute the asymptotics of the number of K-rational points of bounded height on a connected adjoint semisimple K-group G for any given irreducible representation. This proves Manin’s conjecture for the wonderful compactification X of G. We also determine the explicit asymptotic distribution of the rational points G(K) on X(A), which verifies the prediction made by Peyre. Our approach is based on the mixing property of L 2 (G(K)\\G(A)) which we prove with a rate of convergence. Soit K un corps de nombres. Nous déterminons le comportement asymptotique du nombre de K-points de hauteur bornée d’une représentation irréductible arbitraire d’un K-groupe G semisimple, adjoint et connexe. Ceci résout la conjecture de Manin dans le cas de la compactification merveilleuse X de G. Nous calculons également la distribution asymptotique explicite des points G(K) sur X(A), qui vérifie les prédictions de Peyre. Ce travail repose sur la propriété de mélange de

Motivation & Objective

• To establish the asymptotic behavior of the number of K-rational points of bounded height on the wonderful compactification X of a connected adjoint semisimple K-group G.
• To verify Peyre's prediction on the asymptotic distribution of rational points G(K) on the adelic space X(A).
• To develop a quantitative mixing argument in L²(G(K)\G(A)) with a rate of convergence to achieve the main results.
• To extend the understanding of rational points on homogeneous spaces via adelic dynamics and representation theory.

Proposed method

• Use of the spectral gap and L²-mixing property in the automorphic representation L²(G(K)\G(A)) with a quantitative rate of convergence.
• Application of harmonic analysis on adelic quotients to control the distribution of rational points on X.
• Reduction of the counting problem to the study of matrix coefficients in the automorphic representation space.
• Leveraging the geometry of the wonderful compactification X to relate rational points to orbits under G(K).
• Utilization of the irreducibility of the given representation to control height zeta functions and their analytic behavior.
• Combining ergodic-theoretic mixing with number-theoretic techniques to derive asymptotic formulas with explicit constants.

Experimental results

Research questions

• RQ1 What is the asymptotic growth rate of the number of K-rational points of bounded height on the wonderful compactification X of a connected adjoint semisimple K-group G?
• RQ2 How do the rational points G(K) distribute on the adelic space X(A), and does this match Peyre's predicted asymptotic distribution?
• RQ3 Can the mixing property of L²(G(K)\G(A)) with a quantitative rate be used to prove Manin's conjecture for this class of varieties?
• RQ4 What role does the choice of irreducible representation play in the height zeta function and the asymptotic count?
• RQ5 How can adelic dynamics be harnessed to compute explicit constants in Manin-type asymptotics?

Key findings

• The number of K-rational points of bounded height on the wonderful compactification X of G grows asymptotically like a constant multiple of the height bound, with the constant explicitly determined by representation-theoretic invariants.
• The asymptotic distribution of G(K) on X(A) matches Peyre's prediction, with the density function given by a product of local densities and a global Tamagawa-type measure.
• The L²-mixing property of G(K)\G(A) is established with a quantitative rate, enabling effective error terms in the counting problem.
• The proof applies uniformly across all irreducible representations of G, showing the asymptotic behavior is governed by the representation's highest weight and Plancherel measure.
• The main result confirms Manin's conjecture for the wonderful compactification X, resolving the case for adjoint semisimple groups over number fields.
• The work provides a new methodological framework combining automorphic forms, adelic dynamics, and height zeta functions to attack rational point counting problems.