Publications Mathématiques de Besançon
Algèbre et Théorie des Nombres
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Kristin Lauter; Bianca Viray
Denominators of Igusa class polynomials
Publications mathématiques de Besançon no. 2 (2014), p. 5-29, doi: 10.5802/pmb.6
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Class. Math.: 11Y99, 11G15, 14K22, 14G50
Mots clés: Gross-Zagier’s formula, intersection number, complex multiplication, Igusa class polynomials

Résumé

Dénominateurs des polynômes des classes d’Igusa.

Cet article donne une démonstration directe de la formule explicite du nombre d’intersection $\left(\operatorname{CM}(K). G_1\right)$ sur l’espace des modules de Siegel pour un corps $K$ à multiplication complexe quartique. Cette formule permet de calculer, d’une manière effective, les dénominateurs des polynômes des classes d’Igusa ce qui est utile pour construire des courbes de genre $2$ pour la cryptographie. Cette formule a été démontrée dans l’article [22], avec une forte dépendance, dans la démonstration, d’une formule donnée dans [21] qui généralise la formule de Gross et Zagier. Notre présentation ici est plus transparente et plus adaptée pour écrire un algorithme pour la calculer. Nous donnons aussi des exemples et des applications.

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