# How can we view the local Weil group as a group scheme over $\mathbf Q$?

Let $F$ be a $p$-adic local field with residue field $\kappa$, and $q = |\kappa|$. Then the residue field $\overline{\kappa}$ of $F^{\textrm{ur}}$ is an algebraic closure of $\kappa$. The local Weil group $W_F$ is the group of $\sigma \in \operatorname{Gal}(\overline{F}/F)$ such that $\bar{\sigma}(x) = x^{q^n}$ for some $n \in \mathbf Z$ and all $x \in \overline{\kappa}$. Let $I = \operatorname{Gal}(\overline{F}/F^{\textrm{ur}})$ be the inertia group, which is the kernel of $W_F \rightarrow \operatorname{Gal}(\overline{\kappa}/\kappa)$, and $\Phi \in W_F$ be a geometric Frobenius, which induces the automorphism $x \mapsto x^{q^{-1}}$ on $\overline{\kappa}$. Then $W_F$ is topologically the semidirect product of $I$ and $\langle \Phi \rangle \cong \mathbf Z$.

Here is what John Tate says in his Corvallis article on the Weil group:

I don't understand what is going on here. First, is Tate claiming that there is an identification with the underlying space of $\operatorname{Spec} A_n$ and the elements of $\Phi^n I$? Or is he saying that $W_F$ occurs as the $\mathbb Q$-rational points of that $\mathbb Q$-scheme? Do we lose the topology on $W_F$ when we view it as a group scheme?

The construction will produce $W_F$ both as the spectrum and the set of $\mathbb Q$-valued points - i.e. the set of points is $W_F$, and all the residue fields are $\mathbb Q$.
The topology on $W_F$ is visible in the first perspective, as the topology of the scheme but not directly in the second perspective.
It is easies to verify this facts first for $I$ (i.e. that $I$ is the spectrum of the ring of locally constant functions he constructs, and that all the residue fields are $\mathbb Q$) and then observe that they hold for $W_F$ as it is the infinite disjoint union of countably many copies of $I$.