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

The ideles congruent to one modulo $\mathfrak{m}$ have finite components in $1+{\mathfrak{p}}^{n_{\mathfrak{p}}}$ at primes in the finite modulus and positive components at selected real places. Their image in $C_K$ is the idelic congruence subgroup.