Class field theory for orders of number fields

Gene S. Kopp, Jeffrey C. Lagarias

Published 2022-12-18Version 1

This paper defines a generalized notion of "ray class group" associated to an arbitrary order in a number field together with an arbitrary ray class modulus for that order. It shows existence of ray class fields corresponding to these ray 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 an exact sequence 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 identify each ray class field of an order in terms of the splitting behavior of primes. An appendix extends some structural results of the paper to ray class monoids of the order (which include non-invertible elements).