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nLab > Latest Changes: local Langlands conjecture

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  1.  
    • CommentRowNumber1.
    • CommentAuthorDmitri Pavlov
    • CommentTimeDec 30th 2018
    • (edited Dec 30th 2018)

    This article has a weird claim on top, highlighted in yellow (see the second line):

    Redirected from “local Langlands correspondence”.

    Note: local Langlands conjecture and local Langlands conjecture both redirect for “local Langlands correspondence”.

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeDec 30th 2018

    I think I fixed it, by removing a duplicate of the line

      [[!redirects local Langlands correspondence]]

    diff, v5, current

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeDec 30th 2018

    (well, I made the error message go away. But it remains true that this error message was weird.)

  4. I agree that the error message is confusing! It comes from Instiki. Eventually I will move the handling of redirects out of the old Instiki.

    • CommentRowNumber5.
    • CommentAuthorDavid_Corfield
    • CommentTimeFeb 28th 2021

    Added pointer to

    It is curious that our quest was to understand the local Langlands correspondence in an arithmetic setting, for potentially very ramified representations, and eventually we solved it by relating it to the global Langlands correspondence in a geometric setting, in the everywhere unramified setting.

    diff, v6, current

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeFeb 28th 2021

    You have a lonely “For” above the reference. Probably some comment intended but missing.

    • CommentRowNumber7.
    • CommentAuthorDavid_Corfield
    • CommentTimeFeb 28th 2021

    Oh yes, I was going to find something pithy to say, but then forgot. I’ll add

    For an approach via the Fargues–Fontaine curve…

    We’ve discovered at least that this paper uses the tangent (∞,1)(\infty, 1)-topos to condensed spaces, from this MO answer.

    There’s probably still a world to tap into of condensed differential cohesion, but perhaps that will have to wait.

  8. Changed the description of the correspondence to reflect that the correspondence as initially stated is for GL_n. Added the description for more general reductive groups as stated in Chao Li’s notes from Jack Thorne’s class (also added in the references).

    Anton Hilado

    diff, v10, current

  9. Edited the statement to be more accurate (Weil-Deligne representations have to be F-semisimple and the representations of GL_n(F) have to be irreducible and admissible).

    Anton Hilado

    diff, v10, current

  10. Added some important definitions.

    Anton Hilado

    diff, v11, current

  11. Corrected Weil-Deligne representation.

    Anton Hilado

    diff, v11, current

  12. Corrected the example of GL_1.

    Anton Hilado

    diff, v11, current

    • CommentRowNumber13.
    • CommentAuthorDavidRoberts
    • CommentTimeApr 23rd 2021

    Thanks for making all these edits, Anton! It’s much appreciated.

    • CommentRowNumber14.
    • CommentAuthorAnton Hilado
    • CommentTimeApr 23rd 2021
    Thanks! I was also planning to make a page for the global Langlands correspondence for function fields and write up some things regarding V. Lafforgue's work on the automorphic to Galois direction. Apparently I was told this is not quite "geometric" Langlands yet - maybe to be geometric one needs the formulation involving sheaves, which comes from Grothendieck's sheaf-function correspondence. But I could also add this stuff to the geometric Langlands page, although I'm not quite sure how to go about it, where to insert it, etc. Any advice?
    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeApr 23rd 2021

    Sounds like you want a page global Langlands correspondence. Do you know how to create a new page? Let me know if I should do it.

    • CommentRowNumber16.
    • CommentAuthorUrs
    • CommentTimeApr 23rd 2021

    In the Definition-section, I have taken the liberty of stating again the assumption that GG denotes a reductive group (if that’s the case, please check).

    Have added numbered Definition-environments and some cross-pointers.

    Have hyperlinked technical terms, by enclosing them in double square brackets [[...]]:

    e.g. p-adic field, residue field, class field theory

    also for basic ones (such as inverse image, linear representation, irrep, invariant subspace, subgroup),

    diff, v13, current

    • CommentRowNumber17.
    • CommentAuthorAnton Hilado
    • CommentTimeApr 23rd 2021

    Adding in the case of GL_2 (to be completed later).

    diff, v15, current

    • CommentRowNumber18.
    • CommentAuthorAnton Hilado
    • CommentTimeApr 23rd 2021

    Adding in the case of GL_2 (to be completed later).

    diff, v15, current

    • CommentRowNumber19.
    • CommentAuthorAnton Hilado
    • CommentTimeApr 23rd 2021

    Added the case of GL_2 (reference is to lecture 9 of Kevin Buzzard’s MSRI notes). Intending to explain how the different irreducible admissible representations are obtained later (for example as induced representations for the principal series and special ones.

    diff, v15, current

    • CommentRowNumber20.
    • CommentAuthorAnton Hilado
    • CommentTimeApr 23rd 2021

    Edited some typos, added a placeholder for geometrization (I intend to add in some stuff later).

    diff, v15, current

    • CommentRowNumber21.
    • CommentAuthorAnton Hilado
    • CommentTimeApr 23rd 2021
    Thank you for the further edits! I have made a few more edits today, and might add in a little more later. I think I can try for myself creating a new page for the global Langlands correspondence, but I'll be asking for more help here if I run into some difficulty!

    I'd like to ask as well - would it be better to change the page name from "local Langlands conjecture" to "local Langlands correspondence"?
    • CommentRowNumber22.
    • CommentAuthorAnton Hilado
    • CommentTimeApr 23rd 2021

    Added a section on p-adic and mod p local Langlands.

    diff, v16, current

    • CommentRowNumber23.
    • CommentAuthorUrs
    • CommentTimeApr 23rd 2021

    would it be better to change the page name from “local Langlands conjecture” to “local Langlands correspondence”

    I don’t know. If you are expert on these topics and have reason to prefer one over the other, then you are invited to change it.

    • CommentRowNumber24.
    • CommentAuthorAnton Hilado
    • CommentTimeApr 23rd 2021

    Added principal series and the Bernstein-Zelevinsky theorem.

    diff, v17, current

    • CommentRowNumber25.
    • CommentAuthorAnton Hilado
    • CommentTimeApr 23rd 2021

    Added the statement of the “geometric” local Langlands.

    diff, v20, current

    • CommentRowNumber26.
    • CommentAuthorAnton Hilado
    • CommentTimeApr 24th 2021

    Changed the definition of Bun_G (should be G-torsors and not just vector bundles) and created new pages for “L-parameter” and “moduli stack of L-parameters”.

    diff, v21, current

    • CommentRowNumber27.
    • CommentAuthorAnton Hilado
    • CommentTimeJul 4th 2022

    Added mention of solid abelian groups and condensed mathematics.

    diff, v22, current

    • CommentRowNumber28.
    • CommentAuthorAnton Hilado
    • CommentTimeJul 4th 2022

    Added reference to the work of Vincent Lafforgue. Made mention of Fargues and Scholze’s use of “anima” (infinity groupoids) to construct the spectral action.

    diff, v23, current

    • CommentRowNumber29.
    • CommentAuthorAnton Hilado
    • CommentTimeJul 4th 2022

    Added mention of the work of Anschütz and Le Bras that use the spectral action to show existence of the Hecke eigensheaf.

    diff, v23, current

    • CommentRowNumber30.
    • CommentAuthorAnton Hilado
    • CommentTimeJul 4th 2022

    Added Fargues-Scholze’s notation for the moduli stack of L-parameters.

    diff, v23, current

    • CommentRowNumber31.
    • CommentAuthorAnton Hilado
    • CommentTimeJul 23rd 2023

    Added links to p-adic and mod p local Langlands correspondences, and creating pages.

    diff, v24, current

1 to 31 of 31

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