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

TL;DR

This paper extends the classical theory of ray class groups and fields to arbitrary orders in number fields, defining new structures and establishing their properties and relationships.

Contribution

It introduces a generalized notion of ray class groups and fields for arbitrary orders, including Archimedean data, and provides explicit descriptions and exact sequences relating them.

Findings

Defined ray class groups for arbitrary orders with ray class moduli.

Proved existence of corresponding ray class fields.

Established relationships and exact sequences connecting different orders and moduli.

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.