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

TL;DR

This paper investigates the distribution and existence of rational matrices with a fixed denominator in algebraic groups, using adelic methods to establish equidistribution and local-global principles, with applications to quadratic forms.

Contribution

It introduces a new approach combining adelic mixing to prove equidistribution and a local-global principle for rational matrices with a given denominator.

Findings

Proves equidistribution of denominator n matrices in certain algebraic groups.

Establishes a local-global principle for the existence of rational matrices with fixed denominator.

Applies results to quadratic forms and non simply-connected groups.

Abstract

We study some subsets of rational points in an algebraic groups defined by open conditions on their projection in the finite adeles points. Using adelic mixing we are able to prove an equidistribution's result for the projection of these sets in the real points. 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. Then we study the more complicated case of non simply-connected groups applying it to quadratic forms.