= Solution
For $\alpha=(\alpha_v)\in\mathbb I_K$, the congruence $\alpha\equiv1\pmod{\mathfrak m}$ means
$$
\alpha_\mathfrak p\in1+\mathfrak p^{n_\mathfrak p}\mathcal O_\mathfrak p
\quad(\mathfrak p\mid\mathfrak m_0),
\qquad
\alpha_v>0\quad(v\mid\mathfrak m_\infty).
$$
At finite primes outside the modulus require $\alpha_\mathfrak p\in\mathcal O_\mathfrak p^\times$. These ideles form $\mathbb I_K(\mathfrak m)$. Their image
$$
C_K(\mathfrak m)
=K^\times\mathbb I_K(\mathfrak m)/K^\times
$$
inside the <idèle class group> is the <idelic congruence subgroup>.
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