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Ray class groups and ray class fields for orders of number fields

Gene S. Kopp, Jeffrey C. Lagarias

TL;DR

This work extends class field theory from the maximal order to arbitrary orders ${\mathcal O}$ in a number field $K$ by defining ray class groups $\mathop{\mathrm{Cl}}_{\mathfrak m,\Sigma}(\mathcal O)$ and associated ray class fields $H_{\mathfrak m,\Sigma}^{\mathcal O}$. It develops a robust framework of extension/contraction of ideals across orders, proves exact sequences linking unit groups, principal ideals, and ray class groups under changes of order and modulus, and provides a Takagi-based construction of ray class fields for orders. The results show how order-based fields sit between maximal-order ray class fields and ring class fields, and establish an Artin reciprocity principle for these order-based extensions. The paper also demonstrates, via explicit computations in real quadratic fields, that ray class fields of non-maximal orders can differ substantially from the compositum of maximal-order ray class fields and ring class fields, with applications to SIC-POVMs and CM theory. Overall, it provides a concrete, computable bridge between arithmetic of orders and abelian extensions of $K$, enriching both theory and explicit calculations.

Abstract

This paper contributes to the theory of orders of number fields. This paper defines a notion of "ray class group" associated to an arbitrary order in a number field together with an arbitrary ray class modulus for that order (including Archimedean data), constructed using invertible fractional ideals of the order. It shows existence of "ray class fields" corresponding to the class groups. These ray class groups (resp., ray class fields) specialize to classical ray class groups (resp., fields) of a number field in the case of the maximal order, and they specialize to ring class groups (resp., fields) of orders in the case of trivial modulus. The paper gives exact sequences for simultaneous change of order and change of modulus. As a consequence, we identify the ray class field of an order with a given modulus as a specific subfield of a ray class field of the maximal order with a larger modulus. We also uniquely describe each ray class field of an order in terms of the splitting behavior of primes.

Ray class groups and ray class fields for orders of number fields

TL;DR

This work extends class field theory from the maximal order to arbitrary orders in a number field by defining ray class groups and associated ray class fields . It develops a robust framework of extension/contraction of ideals across orders, proves exact sequences linking unit groups, principal ideals, and ray class groups under changes of order and modulus, and provides a Takagi-based construction of ray class fields for orders. The results show how order-based fields sit between maximal-order ray class fields and ring class fields, and establish an Artin reciprocity principle for these order-based extensions. The paper also demonstrates, via explicit computations in real quadratic fields, that ray class fields of non-maximal orders can differ substantially from the compositum of maximal-order ray class fields and ring class fields, with applications to SIC-POVMs and CM theory. Overall, it provides a concrete, computable bridge between arithmetic of orders and abelian extensions of , enriching both theory and explicit calculations.

Abstract

This paper contributes to the theory of orders of number fields. This paper defines a notion of "ray class group" associated to an arbitrary order in a number field together with an arbitrary ray class modulus for that order (including Archimedean data), constructed using invertible fractional ideals of the order. It shows existence of "ray class fields" corresponding to the class groups. These ray class groups (resp., ray class fields) specialize to classical ray class groups (resp., fields) of a number field in the case of the maximal order, and they specialize to ring class groups (resp., fields) of orders in the case of trivial modulus. The paper gives exact sequences for simultaneous change of order and change of modulus. As a consequence, we identify the ray class field of an order with a given modulus as a specific subfield of a ray class field of the maximal order with a larger modulus. We also uniquely describe each ray class field of an order in terms of the splitting behavior of primes.
Paper Structure (45 sections, 39 theorems, 132 equations, 1 figure)

This paper contains 45 sections, 39 theorems, 132 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background
  3. Ray class groups for orders
  4. Ray class fields for orders and class field theory for orders
  5. Applications
  6. Prior work
  7. Contents of the paper
  8. Notation
  9. Ideals of orders
  10. Integral ideals, prime ideals, and primary ideals
  11. Invertible integral ideals of orders of number fields
  12. Conductors and relative conductors of orders of number fields
  13. Monoids of integral ideals coprime to a fixed modulus $\mathfrak{m}$
  14. Fractional ideals of orders
  15. Fractional ideals for integral domains
  16. ...and 30 more sections

Key Result

Theorem 1.1

Let $K$ be a number field, $\mathcal{O}$ an order of $K$, $\mathfrak{m}$ an ideal of $\mathcal{O}$, and ${\Sigma}$ a (possibly empty) subset of the set of real embeddings of $K$. Then for the level datum $(\mathcal{O}; \mathfrak{m}, {\Sigma})$, there exists a unique abelian Galois extension $H_{\mat

Figures (1)

  • Figure 1: Hasse diagram for $\{\mathfrak{q}_{4}, \mathfrak{q}_4', \mathfrak{q}_{16}, \mathfrak{q}_{16}'\}$.

Theorems & Definitions (131)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1: Coprimality of integral ideals
  • Proposition 2.2: Primary decomposition in dimension 1
  • proof
  • Definition 2.3: Invertible integral ideal
  • Lemma 2.4
  • proof : Proof of (1)
  • proof : Proof of (2)
  • ...and 121 more