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
Keywords: Gross-Zagier’s formula, intersection number, complex multiplication, Igusa class polynomials

Abstract

In [22], the authors proved an explicit formula for the arithmetic intersection number $\left(\operatorname{CM}(K). G_1\right)$ 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.

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