TITLE: geometric Langlands for GL(1)
QUESTION [11 upvotes]: I should tell you from the beggining that I don't know almost anything about what I'm going to ask/write.
Let $X$ be a smooth projective (e.g. elliptic) curve over a finite field $\mathbb{F}_q$. Then (by geometric Langlands for $GL(1)$ I've heard) we have some sort of correspondence between characters of
$\pi_1(X)$ and characters of the group $Pic(X)$. (*)
Here $\pi_1(X)$ is the algebraic fundamental group, i.e. the semidirect product of 
$\hat{\mathbb{Z}}$ and $\pi_1^{et}(\overline{X})$.
The question is: how to prove/see (*)? do you know any good reference for it?

REPLY [2 votes]: You start with a local system $\mathcal L$ on $X$. By taking symmetric powers, you get a local
system $Sym^n \mathcal L$ on $Sym^n X$ for any $n$.  Now if you choose $n\geq g,$
then the natural map $Sym^n X \to Pic^n(X)$ is surjective, and the fibres are all projective spaces, in particular simply connected.  Thus $Sym^n \mathcal L$ descends to a local system on $Pic^n(X)$.  This is essentially the desired correspondence.   [Sorry; after writing this I realized that it is covered by Peter Toth's answer.]