Class Invariants for Quartic CM Fields

On peut définir des invariants de classe pour un corps CM quartique primitif $K$ comme valeurs spéciales de certaines fonctions modulaires de Siegel (ou Hilbert) aux points CM associés à $K$. De telles constructions ont été décrites par de Shalit-Goren et Lauter. Nous donnons des bornes explicites pour les idéaux premiers divisant les dénominateurs de ces nombres algébriques. Cela nous permet, en particulier, de construire des $S$-unités dans certaines extensions abéliennes d’un corps réflexe de $K$, où $S$ est explicitement determiné par $K$, et de borner les nombres premiers apparaissant aux dénominateurs des polynômes de classe d’Igusa qui interviennent dans la construction des courbes CM de genre 2, comme dans la conjecture de Lauter.

One can define class invariants for a quartic primitive CM field $K$ as special values of certain Siegel (or Hilbert) modular functions at CM points corresponding to $K$. Such constructions were given by de Shalit-Goren and Lauter. We provide explicit bounds on the primes appearing in the denominators of these algebraic numbers. This allows us, in particular, to construct $S$-units in certain abelian extensions of a reflex field of $K$, where $S$ is effectively determined by $K$, and to bound the primes appearing in the denominators of the Igusa class polynomials arising in the construction of genus 2 curves with CM, as conjectured by Lauter.

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