Timeline for On a remark of Langlands
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
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| Oct 29, 2020 at 13:00 | comment | added | Waleed Qaisar | Thank you! I would be very interested in any resources you recommend I look at. | |
| Oct 27, 2020 at 23:27 | comment | added | dhy | ...The mention of "geometric Langlands" is a red herring. My suggestion boils down to the observation that, in some ways, arithmetic galois groups and (complex) differential galois groups behave similarly. This has been known at least since 1970. (I don't think Langlands had this in mind for his letter to Weil.) But I would be kind of shocked if Langlands was thinking of something else for the link he mentions to the geometric Langlands program. The moment you start thinking about ramification there you instantly run into the theory of irregular singularities. | |
| Oct 27, 2020 at 23:16 | comment | added | dhy | @WaleedQaisar Well, I am not sure what Langlands is referring to in his letter to Weil. Langlands only says that there are "traces of this topic", and I can imagine various ways to see traces (e.g. paul garrett's comments seem plausible to me) but it is certainly not front and center in his letter. That being said, I would say that the link I previously mentioned is very concrete... | |
| Oct 27, 2020 at 20:12 | comment | added | paul garrett | ... [cont'] but/and Langlands would have been aware of those ideas, whether completely proven or not. And, indeed, as you speculate, I doubt that people had in mind "geometric Langlands" in those days. | |
| Oct 27, 2020 at 20:11 | comment | added | paul garrett | @WaleedQaisar, I cannot claim to truly know, but in those years Harish-Chandra had already been using various ideas about higher-rank/dimension analogues of "ODE with regular singular points", in his work going back to the 1950s. Apparently he left a large, unfinished manuscript on this (dunno whether it reached his collected works... I've tried to buy a copy, but they're out of print?), which in fact did not quite succeed in proving the more-general things he needed for his repn theory of semi-simple real Lie groups... [cont'] | |
| Oct 27, 2020 at 19:53 | comment | added | Waleed Qaisar | @paulgarrett I'm afraid I'm too ignorant to say "what I want to hear", but again I think it's unlikely that this is what Langlands is referring to in the letter to Weil? | |
| Oct 27, 2020 at 19:53 | comment | added | Waleed Qaisar | I apologize for the delay in response. @dhy I think I gathered something similar from talking to people about the geometric Langlands side, but is there a more direct link to the arithmetic side? Langlands says "there are traces of this in the long letter to Weil about the Hecke theory" and I can't imagine this is what he was thinking of. Even with geometric Langlands, does it not seem Langlands is referring to something more concrete, especially considering his views on the geometric theory after the Abel paper (please excuse my naivety if this is not so)? | |
| Oct 13, 2020 at 23:23 | comment | added | dhy | My guess is the comment has rather concrete meaning. The study of irregular singular points of ODE is more or less the study of $\mathcal{D}$-modules on the formal disk. The space of local Langlands parameters in geometric Langlands is a moduli space of $\mathcal{D}$-modules on the formal disk. So the study of singularities of ODE is the geometric counterpart of the study of ramification in the arithmetic setting. | |
| Oct 13, 2020 at 22:54 | history | edited | Waleed Qaisar | CC BY-SA 4.0 |
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| Oct 13, 2020 at 22:33 | comment | added | paul garrett | At the very least, the Casselman (-Milicic?) subrepresentation theorem for real reductive groups, improving Harish-Chandra's subquotient theorem, using an elaboration of Deligne's work on PDE, is an essential "local" result for the modern representation theory of real reductive groups, which, yes, is essential for a high-end viewpoint on automorphic forms, Langlands' programme, and such... Is this the sort of thing you're wanting to hear? | |
| Oct 13, 2020 at 22:28 | history | asked | Waleed Qaisar | CC BY-SA 4.0 |