Timeline for What is the current status of the function fields Langlands conjectures?
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| Jan 14, 2010 at 8:17 | vote | accept | Puraṭci Vinnani | ||
| Jan 4, 2010 at 14:49 | comment | added | David Ben-Zvi | In the geometric (ie categorified function field) setting there is a general formulation of local Langlands for any group (rough version due to Frenkel-Gaitsgory, refined by Gaitsgory-Lurie in a preprint). It is a conjectural equivalence between [homotopical versions of 2-categories of] categories acted on by the loop group (categorified analogs of admissible reps of p-adic groups) and categories sheafifed over the stack of dual-group local systems over the punctured disc (analog of Galois representations). It is fixed more canonically by a kind of compatibility with Satake parameters.. | |
| Jan 4, 2010 at 9:17 | comment | added | Kevin Buzzard | My understanding is that away from GL_n (where packets have size 1) it's still in some sense an open question to even formulate a local Langlands conjecture rigorously. Matt/David: feel free to correct me if I'm wrong! I thought that the conjecture's current form was "there is a canonical bijection between (packets of representation-theoretic objects) and (certain Galois-theoretic objects)" but if one doesn't have a true definition of "canonical" (and do we?) then what next? For GL_n we have epsilon factors of pairs but we have nothing like this for general G, right? | |
| Jan 4, 2010 at 5:09 | history | edited | Emerton | CC BY-SA 2.5 |
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| Jan 3, 2010 at 19:41 | comment | added | Emerton | Thanks BZ. I was hoping that you would clarify this aspect of things, about which I'm always unsure! I don't know how much one should distinguish between formal structure and more precise results in this context, since one is short of general theorems in the ``classical Langlands'' function field picture, just as much as in the geometric Langlands picture. | |
| Jan 3, 2010 at 19:15 | comment | added | David Ben-Zvi | Of course Matthew is right in general, but I wanted to add that there are general conjectures in geometric Langlands with arbitrary ramification - starting in general terms in the work of Frenkel-Gaitsgory and developed further in the work in progress of Gaitsgory-Lurie. The extended topological field theory picture of geometric Langlands includes (and one could say requires) arbitrary ramification. But other than formal structure again Matthew is right, most of the understanding is in the tamely ramified case. | |
| Jan 3, 2010 at 18:52 | comment | added | t3suji | Drinfeld proved functional Langlands for $GL(n)$ when $n=2$, which is clearly less general than Lafforgues $GL(n)$ for any $n$. | |
| Jan 3, 2010 at 18:10 | comment | added | Anweshi | A comment: It is mentioned in pages about Drinfel'd that he proved the functional field Langlands. Lafforgue is perhaps more general? | |
| Jan 3, 2010 at 16:53 | history | answered | Emerton | CC BY-SA 2.5 |