Modulus of a number field

A modulus is a formal product $\mathfrak{m}={\mathfrak{m}}_0{\mathfrak{m}}_{\infty }$, where ${\mathfrak{m}}_0$ is a nonzero integral ideal and ${\mathfrak{m}}_{\infty }$ is a product of distinct real places.

The ray class group modulo $\mathfrak{m}$ is $I_K(\mathfrak{m})/P_K(\mathfrak{m})$, where the numerator consists of fractional ideals prime to ${\mathfrak{m}}_0$ and the denominator consists of $(\alpha )$ with $\alpha \equiv 1$ at the finite modulus and positive at every real place in ${\mathfrak{m}}_{\infty }$.

If $U_{\mathfrak{m},1}$ consists of units congruent to one modulo ${\mathfrak{m}}_0$ and positive at the selected real places, then