Paramétrisation de structures algébriques et densité de discriminants

Belabas, Karim

Séminaire Bourbaki : volume 2003/2004, exposés 924-937, Astérisque, no. 299 (2005), Exposé no. 935, pp. 267-299.

La composition de Gauss donne une structure de groupe aux orbites de formes quadratiques binaires entières de discriminant $D$, sous l’action de ${\mathrm{SL}}_2$ par changement de variable, essentiellement le groupe des classes de l’ordre quadratique de discriminant $D$. Les domaines fondamentaux associés permettent calculs explicites et évaluation d’ordres moyens. Je présenterai les lois de composition supérieures découvertes par M. Bhargava à partir de la classification des espaces vectoriels préhomogènes réguliers, ainsi que les résultats de densité qu’il obtient ou conjecture, en particulier sur les discriminants de corps de nombres.

Gauss composition yields a group structure on the orbits of integer binary quadratic forms of discriminant $D$, modulo the natural ${\mathrm{SL}}_2$ action. In essence, it is the class group of the quadratic order of discriminant $D$. Associated fundamental domains allow explicit computations and asymptotic evaluation of average orders. I shall present the higher composition laws discovered by M. Bhargava, their roots in the theory of regular prehomogeneous vector spaces, as well as the density results he obtains or conjectures, in particular concerning discriminants of algebraic number fields.

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