Manin's and Peyre's conjectures on rational points and adelic mixing

2006/01/06 by Alex Gorodnik, Gorodnik, Alex, François Maucourant +3 · 2 citations
Mathematics · #11D45 #11F70 #11G50 #14G25 #22E55 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.math/0601127

openalex publication_date 2006/01/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

Let X be the wonderful compactification of a connected adjoint semisimple group G defined over a number field K. We prove Manin's conjecture on the asymptotic (as T→ ∞) of the number of K-rational points of X of height less than T, and give an explicit construction of a measure on X(A), generalizing Peyre's measure, which describes the asymptotic distribution of the rational points G(K) on X(A). Our approach is based on the mixing property of L²(G(K)\G(A)) which we obtain with a rate of convergence.

Citations

• Existence et equidistribution des matrices de denominateur n dans les groupes unitaires et orthogonaux

Cited by

• M-points of bounded height
• Effective equidistribution, arithmetic purity of strong approximation, and geometric sieve for affine quadrics

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