Idelic congruence subgroup ($C_K(\mathfrak{m})$)

For a modulus of a number field $\mathfrak{m}$, form a subgroup $U(\mathfrak{m})$ of the idele group as follows. At a finite prime dividing ${\mathfrak{m}}_0$ with exponent $n$, use $1+{\mathfrak{p}}^n{\mathcal{O}}_{K_{\mathfrak{p}}}$; at every other finite prime, use ${\mathcal{O}}_{K_{\mathfrak{p}}}^{\times }$. At a real place selected by ${\mathfrak{m}}_{\infty }$, use ${\mathbb{R}}_{>0}$, and at every other infinite place use the entire multiplicative completion.

The image of $U(\mathfrak{m})$ in the idèle class group $C_K$ is the idelic congruence subgroup $C_K(\mathfrak{m})$. Its quotient is the ray class group. The unit requirement outside the modulus is essential: allowing arbitrary valuations at those primes would discard ideal-class information. Multiplication by a principal idele accounts for changing a global generator while preserving its ideal class.