Sur les minima des formes hamiltoniennes binaires définies positives
Publications mathématiques de Besançon. Algèbre et théorie des nombres (2020), pp. 5-25.

Étant donné un ordre maximal 𝒪 d’une algèbre de quaternions rationnelle définie A de discriminant D A , nous montrons que le minimum des formes hamiltoniennes binaires sur 𝒪, définies positives et de discriminant -1, est D A . Lorsque la différente de 𝒪 est principale, nous explicitons une forme atteignant cette valeur, et lorsque 𝒪 est principal, nous donnons la liste exacte des formes atteignant cette valeur. Nous donnons des critères et des algorithmes pour déterminer quand la différente de 𝒪 est principale.

Let A be a definite quaternion algebra over , with discriminant D A , and 𝒪 a maximal order of A. We show that the minimum of the positive definite Hamiltonian binary forms over 𝒪 with discrimiminant -1 is D A . When the different of 𝒪 is principal, we provide an explicit form representing this minimum, and when 𝒪 is principal, we give the list of the equivalence classes of all such forms. We also give criteria and algorithms to determine when the different of 𝒪 is principal.

Publié le :
DOI : https://doi.org/10.5802/pmb.39
Classification : 11E39,  11R52,  11L05,  16H20,  11E20
Mots clés : Quaternion algebra, binary Hamiltonian form, maximal order, Euclidean lattice
@article{PMB_2020____5_0,
     author = {Ga\"etan Chenevier and Fr\'ed\'eric Paulin},
     title = {Sur les minima des formes hamiltoniennes binaires d\'efinies positives},
     journal = {Publications math\'ematiques de Besan\c{c}on. Alg\`ebre et th\'eorie des nombres},
     pages = {5--25},
     publisher = {Presses universitaires de Franche-Comt\'e},
     year = {2020},
     doi = {10.5802/pmb.39},
     language = {fr},
     url = {https://pmb.centre-mersenne.org/articles/10.5802/pmb.39/}
}
Gaëtan Chenevier; Frédéric Paulin. Sur les minima des formes hamiltoniennes binaires définies positives. Publications mathématiques de Besançon. Algèbre et théorie des nombres (2020), pp. 5-25. doi : 10.5802/pmb.39. https://pmb.centre-mersenne.org/articles/10.5802/pmb.39/

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