Ray class monoids for orders of number fields

Abstract

We define and study six ray class monoids associated to a level datum $(\mathcal{O};\mathfrak{m},\Sigma )$, where $\mathcal{O}$ is an order in a number field $K$, $\mathfrak{m}$ is an $\mathcal{O}$-ideal, and $\Sigma$ is a set of real places of $K$. The monoids extend the ray class group (and the ring class group) by including non-invertible classes consisting of non-invertible ideals and ideals not coprime to the modulus. We prove fundamental results on the structure of these ray class monoids.

20 pages; comments welcome!