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There is a very nice geometric proof of Deligne for the Artin Reciprocity in the geometric setting, namely for a smooth, projective, geometrically irreducible curve $C$ over a finite field $\mathbb{F}_{q}$, with function field $K=k(C)$, and idele group $\mathbb{I}_{K}:=\prod^{'}_{p\in|C|}K^{*}_{p}$ there is a one-to-one correspondence between the finite quotients of the double quotient space $k(C)^{*}\backslash\mathbb{I}_{K}/\prod_{p \in |C|}\widehat{\mathcal{O}_{p}^{*}}$ (which is isomorphic to $Pic_{C}(\mathbb{F}_{q})$) and the finite quotients of $\pi^{ab}_{1}(C)$.

Now on the other hand the Artin Reciprocity Law for function fields states (e.g. in Artin-Tate: Class field theory) that the group $\mathbb{I}^{0}_{K}/K^{*}$ of norm 1 idele classes is isomorphic via the Reciprocity map to $Gal(\bar{K}^{ab}/K\bar{k})$.

My questions would be:

  1. These two statements seem to me first as different statements, don´t they?

  2. If we put aside Deligne´s geometric proof for the geometric statement (not seriously and not so for long :-)) then how could one prove the geometric statement using the "number theoretic" Reciprocity Law for function fields?

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up vote 5 down vote accepted

They are different statements. What Deligne proves is the unramified case, i.e. the description of abelian extensions of $K$ unramified everywhere. If you could extend his argument to affine curves then you could possibly prove Artin reciprocity by his method. Going the other way should not be difficult. Have you looked at Serre's book "Groupes algebriques et corps de classes"?

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yes, i am trying now to prove the ramified case, namely for a finite subset S of C i consider 1-dimensional representations of Pic_C,S:= isomorphism classes of line bundles on C with fixed isomorphisms of the stalks at the points in S. On the other hand there is the 1-dimensional representations of the tame fundamental group of the open U:=C-S. But i do not understand, how could i then prove with it the number theoretic Artin Reciprocity? (I think this you mean by the other how does it go?) – Peter Toth Feb 10 '11 at 6:16
I am not sure if just the statement you wrote about the Galois group of $K^{ab}$ is enough but the standard description of the abelian extensions of $K$ of "modulus m", that is with bounded ramification, should do it. It's all is Serre's book. – Felipe Voloch Feb 10 '11 at 6:54
ok, so as i understand for a modulus m we can define the generalized jacobian J_m (which in my notation is basically the Pic^0_C,m and then every abelian, etale covering of U=C-m comes from an isogeny over J_m. So my question transforms to: how can one relate these isogenies to 1-dimensional representations? – Peter Toth Feb 10 '11 at 9:01

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