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Méthodes explicites pour les groupes arithmétiques

Abstract : Central simple algebras have many applications in number theory, but their algorithmic theory is not yet fully developed. I present algorithms to compute effectively with central simple algebras that are both faster and more general than existing ones. Some of these algorithms have proven complexity estimates, a new contribution in this area; others rely on heuristic assumptions but perform very efficiently in practice.Precisely, I consider the following problems: computation of the unit group of an order and principal ideal problem. I start by studying the diameter of fundamental domains of some unit groups using representation theory. Then I describe an algorithm with proved complexity for computing generators and a presentation of the unit group of a maximal order in a division algebra, and then an efficient algorithm that also computes a fundamental domain in the case where the unit group is a Kleinian group. Similarly, I present an algorithm with proved complexity that decides whether an ideal of such an order is principal and that computes a generator when it is. Then I describe a heuristically subexponential algorithm that solves the same problem in indefinite quaternion algebras.
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Aurel Regis Page. Méthodes explicites pour les groupes arithmétiques. Mathématiques générales [math.GM]. Université de Bordeaux, 2014. Français. ⟨NNT : 2014BORD0117⟩. ⟨tel-01111509v2⟩

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