Séance Séminaire

Séminaire Gaston Darboux

Applications of Cartan actions; Littlewood's conjecture, Representatives of ideal classes

Ratner's theorems on unipotent actions (and their generalizations by Margulis and Tomanov resp. Ratner) on locally homogeneous spaces have found many applications to number theory. A class of actions less understood are actions of maximal tori -- here there is a fundamental difference between a rank one and rank two situation. Partial progress towards a classification of invariant measures in the rank two situation by Katok, Lindenstrauss, and myself has lead to a partial result on Littlewood's conjecture in the theory of Diophantine approximation. Other applications are work in progress: Minkowski proved that any ideal class in a number field has representative of norm less than $O(\sqrt D)$ (where D is the discriminant of the field). For totally real fields of degree greater 3 this can (conjecturably) be improved to a $o(\sqrt D)$.