Existence et équidistribution des matrices de dénominateur n dans les groupes unitaires et orthogonaux

Abstract

Let G be a simply-connected ℚ-quasisimple and ℝ-anisotropic algebraic ℚ-group. Let 𝔸 f be the finite part of the adèles 𝔸 of ℚ. Let (H n ) be a sequence of bounded subsets of G(𝔸 f ) which are bi-invariant by a compact open subgroup of G(𝔸 f ). Let Γ n be the projection in G(ℝ) of the sets G(ℚ)∩(G(ℝ)×H n ). Suppose that the volume of the compact subsets G(ℝ)×H n tends to ∞ with n. We prove the equidistribution in G(ℝ) of the Γ n with respect to the Haar probability on G(ℝ). The strategy is to use a mixing result for the action of G(𝔸) on the space L 2 (G(𝔸)/G(ℚ)). As an application, we study the existence and the repartition of rational unitary matrices having a given denominator. We prove a local-global principle for this problem and the equirepartition of the sets of denominator n-matrices when they are not empty. We then study the more complicated case of non simply-connected groups applying it to quadratic forms.

Locations

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