Schertz style class invariants for higher degree CM fields

Mathematics of Computation, vol. 95, pp. 2595–2636

Abstract

Special values of Siegel modular functions for Sp 2 g ⁡ ( Z ) \operatorname {Sp}_{2g}(\mathbf {Z}) generate class fields of complex multiplication (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 Γ 0 ( N ) ⊆ Sp 2 g ⁡ ( Z ) \Gamma ^0 (N)\subseteq \operatorname {Sp}_{2g}(\mathbf {Z}) yield class invariants under some splitting conditions on N 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 N N -system . Such a system consists of quadratic polynomials with coefficients in the real-quadratic subfield satisfying certain congruence conditions modulo N 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.

Authors 2

1. Andreas Enge

   Centre National de la Recherche Scientifique · Université de Bordeaux · Institut national de recherche en sciences et technologies du numérique · CANARI: Analyse cryptographique et arithmétique

   Affiliation as printed

   INRIA, Université de Bordeaux, CNRS, CANARI, 33400 Talence, France

   CANARI - Analyse cryptographique et arithmétique (200, avenue de la Vieille Tour 33405 Talence cedex - France)

   LFANT - Lithe and fast algorithmic number theory (200, avenue de la Vieille Tour 33405 Talence cedex - France)

2. Marco Streng Aachen

   Leiden University

   Affiliation as printed

   Universiteit Leiden, Postbus 9512, 2300 RA Leiden, The Netherlands

   Universiteit Leiden = Leiden University (Leiden University | 2300 RA Leiden The Netherlands - Netherlands)