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Schertz style class invariants for higher degree CM fields

Andreas Enge, Marco Streng

TL;DR

The paper extends Schertz's class invariant framework from genus-1 to Siegel modular functions of higher dimension, enabling the construction of CM invariants with substantially smaller class polynomials. It introduces a general sufficient criterion (via a quadratic form with coefficients in the real quadratic subfield and a level-$N$ Siegel function invariant under $\Gamma^0(N)$) to produce class invariants, and proves that $N$-systems capture all Galois conjugates efficiently. It also develops criteria for when CM values lie in real subfields, either through ramification patterns or Fricke involutions, and provides concrete families of level-$N$ functions (from Igusa invariants and theta products) with practical computations. The numerical examples demonstrate tangible reductions in polynomial height and illustrate reproducible pipelines for constructing Hilbert and ramified-level CM fields, with clear implications for explicit CM abelian varieties and explicit class field theory in higher genus. Overall, the work advances algorithmic CM theory in dimension $g\ge 2$ and broadens the toolkit for explicit CM constructions.

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 $Γ^0 (N)\subseteq \operatorname{Sp}(\mathbb{Z})$ 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.

Schertz style class invariants for higher degree CM fields

TL;DR

The paper extends Schertz's class invariant framework from genus-1 to Siegel modular functions of higher dimension, enabling the construction of CM invariants with substantially smaller class polynomials. It introduces a general sufficient criterion (via a quadratic form with coefficients in the real quadratic subfield and a level- Siegel function invariant under ) to produce class invariants, and proves that -systems capture all Galois conjugates efficiently. It also develops criteria for when CM values lie in real subfields, either through ramification patterns or Fricke involutions, and provides concrete families of level- functions (from Igusa invariants and theta products) with practical computations. The numerical examples demonstrate tangible reductions in polynomial height and illustrate reproducible pipelines for constructing Hilbert and ramified-level CM fields, with clear implications for explicit CM abelian varieties and explicit class field theory in higher genus. Overall, the work advances algorithmic CM theory in dimension and broadens the toolkit for explicit CM constructions.

Abstract

Special values of Siegel modular functions for 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 yield class invariants under some splitting conditions on , 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{-system}. Such a system consists of quadratic polynomials with coefficients in the real-quadratic subfield satisfying certain congruence conditions modulo . 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.

Paper Structure

This paper contains 25 sections, 26 theorems, 109 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. The notion of a class invariant
  3. Siegel modular functions
  4. CM theory
  5. Class fields
  6. Class invariants from functions for Gamma0 (N)
  7. Polarised ideal classes and symplectic bases
  8. Quadratic polynomials and polarised ideal classes
  9. Class invariants
  10. Existence of quadratic polynomials with N dividing C
  11. N-systems
  12. Galois conjugates from N-systems
  13. Existence and computation of N-systems
  14. Complex conjugation
  15. Real class polynomials for ramified levels
  16. ...and 10 more sections

Key Result

Theorem 1.1

Let $\mathcal{O}$ be an order in a CM field $K$ of degree $2g$ that is closed under complex conjugation and contains $\mathcal{O}_{K_0}$, and assume that the different of $K_0$ is principal. Let $\Phi$ be a primitive CM type such that there exists a polarised ideal class for $(\mathcal{O}, \Phi)$. L

Theorems & Definitions (62)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Corollary 3.3
  • ...and 52 more