Denominators of Igusa class polynomials

Lauter, Kristin; Viray, Bianca

Publications mathématiques de Besançon. Algèbre et théorie des nombres, no. 2 (2014), pp. 5-29.

Dénominateurs des polynômes des classes d’Igusa.

Cet article donne une démonstration directe de la formule explicite du nombre d’intersection $\left(CM\left(K\right).G_1\right)$ 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.

In [22], the authors proved an explicit formula for the arithmetic intersection number $\left(CM\left(K\right).G_1\right)$ on the Siegel moduli space of abelian surfaces, under some assumptions on the quartic CM field $K$. These intersection numbers allow one to compute the denominators of Igusa class polynomials, which has important applications to the construction of genus $2$ curves for use in cryptography. One of the main tools in the proof was a previous result of the authors [21] generalizing the singular moduli formula of Gross and Zagier. The current paper combines the arguments of [21, 22] and presents a direct proof of the main arithmetic intersection formula. We focus on providing a stream-lined account of the proof such that the algorithm for implementation is clear, and we give applications and examples of the formula in corner cases.

Classification : 11Y99, 11G15, 14K22, 14G50
Mots clés : Gross-Zagier’s formula, intersection number, complex multiplication, Igusa class polynomials

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