Parametrization of algebraic structures and density of discriminants

Karim Belabas

Séminaire Bourbaki (2003-2004)

  • Volume: 46, page 267-300
  • ISSN: 0303-1179

Abstract

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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.

How to cite

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Belabas, Karim. "Paramétrisation de structures algébriques et densité de discriminants." Séminaire Bourbaki 46 (2003-2004): 267-300. <http://eudml.org/doc/252158>.

@article{Belabas2003-2004,
abstract = {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.},
author = {Belabas, Karim},
journal = {Séminaire Bourbaki},
keywords = {prehomogeneous vector space; density; discriminant; composition laws; number rings},
language = {fre},
pages = {267-300},
publisher = {Association des amis de Nicolas Bourbaki, Société mathématique de France},
title = {Paramétrisation de structures algébriques et densité de discriminants},
url = {http://eudml.org/doc/252158},
volume = {46},
year = {2003-2004},
}

TY - JOUR
AU - Belabas, Karim
TI - Paramétrisation de structures algébriques et densité de discriminants
JO - Séminaire Bourbaki
PY - 2003-2004
PB - Association des amis de Nicolas Bourbaki, Société mathématique de France
VL - 46
SP - 267
EP - 300
AB - 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.
LA - fre
KW - prehomogeneous vector space; density; discriminant; composition laws; number rings
UR - http://eudml.org/doc/252158
ER -

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