Gauss composition yields a group structure on the orbits of integer binary quadratic forms of discriminant , modulo the natural action. In essence, it is the class group of the quadratic order of discriminant . 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 , sous l’action de par changement de variable, essentiellement le groupe des classes de l’ordre quadratique de discriminant . 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.
Mot clés : espace vectoriel préhomogène, densité, discriminant, lois de composition, anneaux de nombres
Keywords: prehomogeneous vector space, density, discriminant, composition laws, number rings
@incollection{SB_2003-2004__46__267_0, author = {Belabas, Karim}, title = {Param\'etrisation de structures alg\'ebriques et densit\'e de discriminants}, booktitle = {S\'eminaire Bourbaki : volume 2003/2004, expos\'es 924-937}, series = {Ast\'erisque}, note = {talk:935}, pages = {267--299}, publisher = {Association des amis de Nicolas Bourbaki, Soci\'et\'e math\'ematique de France}, address = {Paris}, number = {299}, year = {2005}, mrnumber = {2167210}, zbl = {1090.11066}, language = {fr}, url = {http://www.numdam.org/item/SB_2003-2004__46__267_0/} }
TY - CHAP AU - Belabas, Karim TI - Paramétrisation de structures algébriques et densité de discriminants BT - Séminaire Bourbaki : volume 2003/2004, exposés 924-937 AU - Collectif T3 - Astérisque N1 - talk:935 PY - 2005 SP - 267 EP - 299 IS - 299 PB - Association des amis de Nicolas Bourbaki, Société mathématique de France PP - Paris UR - http://www.numdam.org/item/SB_2003-2004__46__267_0/ LA - fr ID - SB_2003-2004__46__267_0 ER -
%0 Book Section %A Belabas, Karim %T Paramétrisation de structures algébriques et densité de discriminants %B Séminaire Bourbaki : volume 2003/2004, exposés 924-937 %A Collectif %S Astérisque %Z talk:935 %D 2005 %P 267-299 %N 299 %I Association des amis de Nicolas Bourbaki, Société mathématique de France %C Paris %U http://www.numdam.org/item/SB_2003-2004__46__267_0/ %G fr %F SB_2003-2004__46__267_0
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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