Paramétrisation de structures algébriques et densité de discriminants  [ Parametrization of algebraic structures and density of discriminants ]
Séminaire Bourbaki : volume 2003/2004, exposés 924-937, Astérisque, no. 299 (2005), Talk no. 935, p. 267-299

Gauss composition yields a group structure on the orbits of integer binary quadratic forms of discriminant D, modulo the natural 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.

La composition de Gauss donne une structure de groupe aux orbites de formes quadratiques binaires entières de discriminant D, sous l’action de 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.

Classification:  11R04,  11R45,  11R29
Keywords: prehomogeneous vector space, density, discriminant, composition laws, number rings
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     title = {Param\'etrisation de structures alg\'ebriques et densit\'e de discriminants},
     booktitle = {S\'eminaire Bourbaki : volume 2003/2004, expos\'es 924-937},
     author = {Collectif},
     series = {Ast\'erisque},
     publisher = {Association des amis de Nicolas Bourbaki, Soci\'et\'e math\'ematique de France},
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Belabas, Karim. Paramétrisation de structures algébriques et densité de discriminants, in Séminaire Bourbaki : volume 2003/2004, exposés 924-937, Astérisque, no. 299 (2005), Talk no. 935, pp. 267-299. http://www.numdam.org/item/SB_2003-2004__46__267_0/

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