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I'm currently working through Frenkel's beautiful paper: http://arxiv.org/PS_cache/hep-th/pdf/0512/0512172v1.pdf. I'm looking for a good example of a projective curve to get my hands dirty, and go through the general constructions that Frenkel shows there and try to do them manually for this example of a curve. Are there any good instructive examples for doing this? (Or does it always get out of hand very quickly?)

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Unfortunately I don't think geometric Langlands is very easy on any curve. The only curve where the objects are readily accessible is $P^1$, but even there the general statement is kind of tricky (see Lafforgue's note here). I would look at Frenkel's writings on the Gaudin model, which is a concrete illustration of the Beilinson-Drinfeld-Feigin-Frenkel approach to geometric Langlands for $P^1$ with several punctures. Also Arinkin and Lysenko worked out explicitly a case of geometric Langlands (in a stronger sense) on $P^1$ minus 4 points -- see the first four papers on a mathscinet search for Arinkin. So the answer is try $P^1$ with some punctures, but don't be surprised if things are rather tricky already there.

(I also think geometric Langlands on an elliptic curve should be accessible, but as far as I know it hasn't been worked out very explicitly.)

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Thank you very much for taking the time to answer my question. I'd love to try and see what can be said for geometric Langlands for elliptic curves. By saying it hasn't been worked out ''very explicitly", do you mean that no work has been done at all in this direction, or that there is some paper with partial results in this direction? – rajamanikkam Dec 19 at 6:23
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 In genus one there's a very explicit description of an open locus in the moduli stack of bundles (namely the Atiyah description of moduli of semistable vector bundles and its variants). This can be seen easily eg from the Fourier-Mukai transform. One can then try to write down D-modules explicitly - this is closely related to elliptic Calogero-Moser systems. One paper I like about this is by Nevins, 0804.4170. There are also many related arXiv papers by B. Enriquez (with obvious titles), in particular writing down quantum Hitchin hamiltonians explicitly. But there's still a long way to go.. – David Ben-Zvi Dec 19 at 21:53
Thank you, that was extremely helpful. I'll keep reading Frenkel's paper, and those you have suggested, and try to digest what you have said. – rajamanikkam Dec 20 at 13:15

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