Weil group - Knowlpedia

Let $F$ be a nonarchimedean local field, let $\Gamma_F=\operatorname{Gal}(F^{\mathrm s}/F)$ be its absolute Galois group, and let $I_F$ be its inertia subgroup. The Weil group $W_F$ is the inverse image of $\mathbb Z\subset\widehat{\mathbb Z}$ under

\Gamma_F\longrightarrow \operatorname{Gal}(\overline{k}_F/k_F)\simeq\widehat{\mathbb Z}.

It is topologized so that $I_F$ is an open subgroup with its profinite topology and $W_F/I_F\simeq\mathbb Z$ is discrete. Thus $W_F$ is dense in $\Gamma_F$ , but its finer locally profinite topology allows continuous complex representations with noncompact Frobenius image.

Frobenius convention

Fix geometric Frobenius $\operatorname{Fr}_F$ , inverse to the map $x\mapsto x^{q_F}$ on the residue-field closure. Then $W_F/I_F$ is generated by $\operatorname{Fr}_F$ , and the normalized absolute value on $W_F$ is

|w|_F=q_F^{-n} \quad\text{if}\quad wI_F=\operatorname{Fr}_F^n I_F.

Using arithmetic Frobenius reverses $n$ ; formulas involving monodromy or local factors must state which convention is in force.

Archimedean Weil groups

For the archimedean local fields,

W_{\mathbb C}=\mathbb C^\times, \qquad W_{\mathbb R}=\mathbb C^\times\sqcup j\mathbb C^\times,

where $j^2=-1$ and $jzj^{-1}=\overline z$ . The quotient $W_{\mathbb R}/\mathbb C^\times$ is $\operatorname{Gal}(\mathbb C/\mathbb R)$ .

Global warning

A global Weil group is not obtained by simply repeating the inverse-image construction above. It is an extension built from global class formations and maps to the local Weil groups. This knowl uses $W_F$ locally unless the word “global” is explicit.

For local fields, abelianization is identified with $F^\times$ by local class field theory.

References

John Tate, “Number theoretic background,” in Automorphic Forms, Representations and $L$ -Functions, Proceedings of Symposia in Pure Mathematics 33, part 2, 1979.
Jayce R. Getz, An Introduction to Automorphic Representations, §§10.1–10.2. Author notes.