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## geometric Langlands for GL(1)

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?

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This is a special case of class field theory for global function fields, which identifies the topological abelianization of $\pi_1(X)$ with the idelic description of ${\rm{Pic}}(X)$. It logically (and historically) comes much before anything with Langlands' name attached to it. – BCnrd Oct 12 2010 at 16:24
A classical reference is Serre's Algebraic groups and class fields. – S. Carnahan Oct 12 2010 at 16:42
@Scott could you please be more precise? A chapter, a section or a theorem? My knowledge about class field theory is very close to zero and I'm kinda lost in this book... – Dragos Fratila Oct 13 2010 at 13:02

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.

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David Ben Zvi also gave an excellent lecture on this topic at MSRI. MSRI's streaming video seems to be down for the moment, but you might want to watch it when it comes back up. – David Speyer Oct 13 2010 at 0:26
I know about Frenkel's notes but they are not helping me too much at the moment. What I would like is an "elementary" proof so I can see the correspondance concretely: say if I take a character on the left what does it look like on the right. – Dragos Fratila Oct 14 2010 at 14:27

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.

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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.]

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