In Serre's "Abelian $\ell$-adic representations and Elliptic curves", in order to define the set $S_{\mathfrak{m}}$ he defines of $U_{v,\mathfrak{m}}$ as follows:
$$U_{v,\mathfrak{m}}=\left\{\begin{array}{ll} \text{The connected component of } K_v^{\times} & \text{ if } v \in \Sigma^{\infty}\\ U_v & \text { if }v \in \Sigma \backslash S \\ \text{Set of elements }x \in U_v \text{ s.t. }v(1-x)\ge m_v & \text { if } v \in S \end{array}\right. $$
where,
- $K$ is a number field,
- $\Sigma$ is the set of non-archimedean places of $K$,
- $\Sigma^{\infty}$ the set of archimedean places of $K$,
- $K_v$ the completion of $K$ w.r.t $v \in \overline{\Sigma}=\Sigma \cup \Sigma^{\infty}$,
- $U_v$ the set of unit of $K_v$,
- $S$ a finite subset of $\overline{\Sigma}$
- $\mathfrak{m}=(m_v)_{v\in S}$ a modulus of support $S$ where $m_v \in \mathbb{Z}_{\ge 1}$
Then, he defines $U_{\mathfrak{m}}=\prod_{v}U_{v,\mathfrak{m}}$ and claims that it is an open subgroup of the idele group $\mathbb{I}$. My questions are as follows:
The components here seem to be groups and not elements of groups. I don't exactly understand how this is an element in $\mathbb{I}$?
Why is this set open in $\mathbb{I}$?
What is the motivation of defining this set like this? This unfortunately seems very arbitrary to me currently. Any references would be appreciated as well, thank you!