Timeline for What are the local Langlands conjectures nowadays, for connected reductive groups over a $p$-adic field?
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5 events
| when toggle format | what | by | license | comment | |
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| Feb 17, 2010 at 1:18 | vote | accept | Kevin Buzzard | ||
| Mar 14, 2021 at 19:35 | |||||
| Feb 12, 2010 at 4:44 | comment | added | David Ben-Zvi | I haven't thought carefully about the abelian case (or others really!), but I imagine it should be an application of the Contou-Carrere self-duality of the Jacobian over the disc - a kind of 2-categorical Fourier-Mukai transform for it - which is also behind geometric CFT.. something to think about! | |
| Feb 10, 2010 at 20:04 | comment | added | Laurent F. | David - Of course I did not mean you expect a geometric classification of supercuspidals, maybe I misspoke, sorry for this. I just wanted to point the big difference between the p-adic and the real case that makes the real case more manageable with the geometric methods. I did not know their is a precise conjecture for geometric Langlands in any depth, thanks for this. Just a question: is it proved for $GL_1$ (by purely local methods, without globalizing to a smooth projective curve) ? Can you link this to Serre's geometric class field theory ? | |
| Feb 10, 2010 at 16:00 | comment | added | David Ben-Zvi | Laurent - thanks for a very enlightening answer! Certainly I wouldn't expect a geometric classification of supercuspidals.. but one can still ask for a geometric (categorical?) description one Bernstein component at a time (as Marty suggests), like we have in the tamely ramified case.. for GL_n I guess this follows from type theory as you explained? There is now a local geometric Langlands program for arbitrary depth.. it's rather formal right now except in some special cases, but there is a precise general conjecture (the tamely ramified case is the theorem of Bezrukavnikov McGerty mentions) | |
| Feb 10, 2010 at 14:45 | history | answered | Laurent F. | CC BY-SA 2.5 |