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:

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

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?