geometric Langlands for GL(1)

Question

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?

Answer 1

The main idea of the proof is the following: the 1-dimensional representations $\{\pi_{1}^{ab}(X) \rightarrow \bar{\mathbb{Q}}_{l}^{*}\}$ are in 1-1 correspondence with 1-dim local systems $L$ on $X$, on the other hand 1-dimensional representations $\{Pic_{X} \rightarrow \bar{\mathbb{Q}}_{l}^{*}\}$ are in 1-1 correspondence with 1-dimensional local systems $A_{L}$ on $Pic_{X}$ together with a rigidification, i.e. a fixed isomorphism $A_{L}|_{0} \cong \bar{\mathbb{Q}}_{l}$ (this is the famous faisceaux-fonctions correspondence of Grothendieck). Now consider the d-symmetric product $X^{(d)}$ of $X$ (which is just the effective divisors of degree $d$ on $X$) which maps in an obvious way to $Pic_{X}^{d}$, the degree $d$ component of the Picard. If there is given a local system $L$ on $X$ then we can produce a local system $L^{(d)}$ on $X^{(d)}$ by having this local system fibres $\bigotimes_{i} Sym^{d_{i}}(L_{x_{i}})$ over a point $\sum_{i} d_{i}x_{i}$. Now by Riemann-Roch if the degree $d$ is greater than $2g(X)-2$ then it follows that this map has fibre over a degree-$d$ line bundle $\mathfrak{L}$ the $d-g(X)$-dimensional projective space $\mathbb{P}(H^{0}(X,\mathfrak{L}))$. As projective spaces are simply connected, it follows that this locally constant sheaf $L^{(d)}$ is actually constant on these fibres, so descends to a local system $A_{L}$ on $Pic_{X}^{d}$. There is also a way to extend these construction to the remaining components using the natural action $X \times Pic \rightarrow Pic$ given by $(x,L) \mapsto L(x)$. It is the idea of the proof, which is actually of Deligne!

As what the references concerns: there is a paper of Laumon: Faisceaux automorphes lies aux series Eisenstein, where he discusses this proof of Deligne. Also here http://www.cims.nyu.edu/~tschinke/publications.html the 3rd book (Mathematisches Institut, Seminars 2003/04, Universitätsverlag Göttingen, (2004) ) from page 145, but it is in german. And also the quoted paper of Frenkel is very good.

Answer 2

If you want to see class field theory written about as if it were an example of Geometric Langlands, try Edward Frenkel's review article.

Answer 3

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.]