# Does local Langlands say anything about the isomorphism class of the absolute Galois group?

I have heard some people claim that the Local Langlands Correspondence over $$\mathbb{Q}_p$$ (when it is known) is a deep theorem about representations of the absolute Galois group of $$\mathbb{Q}_p$$. My impression was that this correspondence is in some sense about the context in which the absolute Galois group of $$\mathbb{Q}_p$$ arises, not about the topological group itself.

Did local Langlands every say anything non-trivial/something that was not known before about the the absolute Galois group of $$\mathbb{Q}_p$$ considered abstractly as a topologically finitely presented topological group?

Basically, you are allowed to refer to the topology and the group structure on this group and thus any invariants depending only on that (e.g. the minimum number of topological generators or the number of conjugacy classes of subgroups of index $$3$$) but you are not allowed to directly refer to varieties, motives, $$L$$-functions (because we only care about the topological group, not about the context where it arises).

An example of how an answer might have looked in an alternate universe: say, the absolute Galois group of $$\mathbb{Q}_p$$ just happened to be finite but people did not know its order and only through local Langlands correspondence were they able to compute it. This is no doubt utter bullshit but I am just indicating the format of a potential answer.

• An analogy: the global Langlands program for $\mathrm{GL}(2)$ says quite a lot about the arithmetic of torsion free congruence subgroups $\Gamma$ of $\mathrm{GL}_2(\mathbf{Z})$, but I don't think it has had very much to say about these groups as abstract groups, since they are, in the end, just free groups. – Puddles the turtle Aug 21 at 15:11