What is the Weil group of a global field $K$?

Question

The question is in the title.

For context, I know some things about the local Weil group. I know that the abelianization is isomorphic to the multiplicative group $K^{\times}$, and I know that it is essentially just the absolute Galois group $G_K$, except that the $\hat{\mathbb{Z}}$ for Frobenius is replaced by a plain $\mathbb{Z}$ to make the group less compact.

There is a corresponding abelianization statement for the global Weil group, namely that its abelianization is approximately the idele class group, maybe doing something with the infinite places. This also suggests some connection with the absolute Galois group, and the paper I’m reading certainly assumes this view.

But the definition (an inverse limit of extensions of Galois groups by ideles given by fundamental classes via CFT) is really mysterious. How should I think of the global Weil group?

Answer 1

In general, the concept of a Weil group arises via the theory of class formations, with the standard introductory reference being Tate's Number theoretic background. In the case that the field is (nonarchimedean, but something similar works for the archimedean case) local, this can be reinterpreted as the definition that you give. However, I'm not aware of any such global reinterpretation.

The general picture is that there is a definition of "a" Weil group relative to a separable algebraic closure $\bar{F}/F$ of a global field $F$. Given this, a Weil group for $\bar{F}/F$ is a topological group $W_F$ equipped with a continuous map $\varphi:W_F\rightarrow Gal(\bar{F}/F)$ with dense image, satisfying various conditions:

• Since $\varphi$ is continuous, for each finite Galois $E/F$ in $\bar{F}$, the pre-image $W_E$ of $Gal(\bar{F}/E)$ is an open subgroup, and since the image is dense there's a canonical isomorphism $W_F/W_E\simeq Gal(E/F)$.
• For each Galois $E/F$, there should be a canonical isomorphism $\Bbb{A}_E^\times/E^\times\simeq W_E^{ab}$, which together with $\varphi$, induces the Artin reciprocity map $\Bbb{A}_E^\times\rightarrow Gal(E/F)^{ab}$.
• These maps all satisfy the usual functorial diagrams from class field theory.
• $W_F$ is canonically isomorphic to $\varprojlim W_F/W_E^c$, where $W_E^c$ denotes the closure of the commutator subgroup of $W_E$.

The groups $W_F/W_E^c$ appeearing in the latter point are often denoted $W_{E/F}$.

Now suppose that one has a construction of such a Weil group. It follows that there's a short exact sequence $$0\rightarrow W_E/W_E^c\rightarrow W_{E/F}\rightarrow W_F/W_E\rightarrow 0.$$ Since $W_E/W_E^c\simeq \Bbb{A}_E^\times/E^\times$ and $W_F/W_E\simeq Gal(E/F)$, this is saying that $W_F$ is an extension of $Gal(E/F)$ by $\Bbb{A}_E^\times/E^\times$; such extensions correspond to classes in $H^2(Gal(E/F),\Bbb{A}_E^\times/E^\times)$. So, assuming the existence of a global Weil group, we get a collection of fundamental classes $\alpha_{E/F}\in H^2(Gal(E/F),\Bbb{A}_E^\times/E^\times)$.

It turns out that these classes are related to eachother in a very nice manner by inflation and restriction. This, together with a natural isomorphism $H^n(Gal(E/F),\Bbb{Z})\rightarrow H^{n+2}(Gal(E/F),\Bbb{A}_E^\times/E^\times)$ induced by taking the cup product with $\alpha_{E/F}$ actually encodes all of the cohomological algebra that goes into class field theory. Moreover, using class field theory one can prove that such a system of fundamental classes exists and is unique, which gives a definition of $W_{E/F}$ for each finite Galois $E/F$, and one then sets $W_F=\varprojlim W_{E/F}$.

So the upshot is that the Weil group is the canonical object which is lurking under the surface of class field theory, which really encodes the relationship between Galois groups of abelian extensions and idele class groups. It's the natural thing whose abelianisation leads to the canonical isomorphism given by class field theory, and which is compatible with all of the various operations that it needs to be.