Schertz style class invariants for higher degree CM fields

Abstract

Special values of Siegel modular functions for $\operatorname{Sp} (\mathbb{Z})$ generate class fields of CM fields. They also yield abelian varieties with a known endomorphism ring. Smaller alternative values of modular functions that lie in the same class fields (class invariants) thus help to speed up the computation of those mathematical objects. We show that modular functions for the subgroup $\Gamma^0 (N)\subseteq \operatorname{Sp}(\mathbb{Z})$ yield class invariants under some splitting conditions on $N$, generalising results due to Schertz from classical modular functions to Siegel modular functions. We show how to obtain all Galois conjugates of a class invariant by evaluating the same modular function in CM period matrices derived from an \emph{$N$-system}. Such a system consists of quadratic polynomials with coefficients in the real-quadratic subfield satisfying certain congruence conditions modulo $N$. We also examine conditions under which the minimal polynomial of a class invariant is real. Examples show that we may obtain class invariants that are much smaller than in previous constructions.