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Geometric class field theory (curves over a finite field) has been generalized to higher dimensional varieties over a finite field (and other arithmetical fields). Some of the key names here are Lang, Serre, Bloch, Kato-Saito...

Question: Does there exist a precise formulation of Langlands's conjectures which constitutes a non-abelian generalization of these results?

Probably relevant: Kazhdan's talk here at the Panel on Open Questions held at Gross's 60th Birthday Conference.

SGP
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    As far as I know, there is no formulation of geometric Langlands for higher dimensional varieties. However, there is a lot of interesting things in this direction discussed here: http://mathoverflow.net/questions/6457/kapranovs-analogies – Puraṭci Vinnani Apr 03 '11 at 00:25
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    Thanks for the link to the other MO question. Ben-Zvi's answer seems to suggest that generalizations of the geometric version seem to be limited by physics (I may be misreading his answer -- I apologize).

    I am wondering about a generalization of the old-fashioned classical (arithmetical) Langlands's conjectures.

     – SGP Apr 03 '11 at 01:31

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From what I remember, Kapranov's paper "Analogies between the Langlands correspondence and topological quantum field theory" has some speculations about this. A wise old friend once called it the most speculative paper he'd ever seen in print.

JBorger
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  • Thanks! very interesting paper..but sadly it does not contain a precise formulation of the conjectural generalization. – SGP Apr 03 '11 at 01:24