Schertz style class invariants for higher degree CM fields

TL;DR

This paper extends the theory of class invariants for CM fields using Siegel modular functions for subgroups, enabling more efficient computation of class fields and abelian varieties with known endomorphism rings.

Contribution

It generalizes Schertz's results to Siegel modular functions for -systems, providing methods to compute all Galois conjugates and conditions for real minimal polynomials.

Findings

Class invariants can be obtained from -systems satisfying certain congruences.

New class invariants are often smaller than previous constructions.

Methods allow explicit computation of Galois conjugates of class invariants.

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.