I'm trying to understand the main theorem of geometric class field theory. Could someone tell me if this example is correct?
Main theorem. Let $K$ be a function field of a curve over a finite field. There is a bijection between unramified $\ell$-adic Galois representations of $G_K := \operatorname{Gal}(K^{sep}/K)$ and $\ell$-adic characters of $K^{\times}\backslash\mathbf{A}_K^{\times}/\mathcal{O}_K^{\times}.$ (Here I mean the ideles modulo the diagonal on the left, and on the right quotiented by the subgroup of ideles which have non-negative valuation at every place.)
I was trying to understand what this means when $K = \mathbb{F}_p(T),$ say. In that case, I think unramified Galois representations are the same as representations of $\operatorname{Gal}(K^{un}/K) = \operatorname{Gal}(\overline{\mathbf{F}_p}(T)/\mathbf{F}_p(T))= \hat{\mathbf{Z}}.$
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On the other side, I believe that I can simplify that double quotient as follows. The places of $K$ are described as follows: there is one place with uniformizer $1/T,$ and then for every monic irreducible polynomial $f(T) \in \mathbb{F}_p[T],$ there is a place with uniformizer $f.$ These are distinct places, and every place is of one of these two forms.
There is a surjective map $\mathbf{A}_K^{\times}\to \bigoplus_{v\in P} \mathbf{Z},$ for $P$ the set of places of $K,$ sending an idele to its valuation at each place. The kernel is precisely $\mathcal{O}_K^{\times},$ and so $$\mathbf{A}_K^{\times}/\mathcal{O}_K^{\times} \cong \bigoplus_{v\in P} \mathbf{Z}.$$
The diagonal embedding of $K^{\times}$ then corresponds to the subgroup of $\bigoplus_{v\in P}\mathbf{Z}$ consisting of elements whose sum is $0.$ In particular, the quotient by the diagonal subgroup leaves us with just $\mathbf{Z}.$
As a sanity check, $\operatorname{Pic}(\mathbb{P}^1) = \mathbf{Z},$ as found in any book on algebraic geometry, and I believe part of the proof of geometric class field goes by identifying the double quotient with the Picard group, so it makes sense we got $\mathbf{Z}$ here.
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So, I think that in this case, it is telling me that continuous homomorphisms $\rho : \hat{\mathbf{Z}} \to \overline{\mathbf{Q}_{\ell}}^{\times}$ are in bijection with continuous homomorphisms $\mathbf{Z}\to \overline{\mathbf{Q}_{\ell}}^{\times}.$
But this surely is incorrect -- the image of a map out of the profinite integers must be compact, but the image out of $\mathbf{Z}$ is not.
So, where have I made a mistake?