Timeline for Relation between motives and geometric Langlands
Current License: CC BY-SA 4.0
15 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 27 at 22:24 | vote | accept | JustLikeNumberTheory | ||
| Mar 26 at 7:22 | comment | added | Alexander Chervov | The cohomology dimension "i" always arises in the same way - $R^i \pi_*$, that is we consider the derived direct image. For example what are the cohomology themselves - these just direct images to the point: $M \to pt$. In the context of the geometric/motivic local system it is described in the paper you mention, see also mathoverflow.net/questions/435745/… So conceptually both things are the same, but technically their might be some differences, that is not so clear for me. | |
| Mar 25 at 15:59 | comment | added | JustLikeNumberTheory | @AlexanderChervov If we have a variety A->X (using X instead of K), then the fundamental group of X naturally acts on the etale cohomology $H^i(A\times \overline{X})$. The action of this fundamental group is enough to define a local system on X. I do not know how to make your D-module definition work; you never make a choice of i, which we need because we want to attach the local system to the motive $h^i(A)$ rather than the variety $A$. | |
| Mar 25 at 9:09 | comment | added | Alexander Chervov | Concerning the pre-last paragraph: "Let X be a (smooth, connected)... ". About: "for A over K " we get "local system E on X". To get that local system - we just consider the direct image of trivial D-module/(=perverse-sheaf) from "A" to "X". Is it correct ? Or I am missing something ? | |
| Mar 24 at 18:10 | comment | added | Alexander Chervov | Thank you, that it is cool ! | |
| Mar 24 at 18:07 | comment | added | JustLikeNumberTheory | By the way, the question you ask seems fairly well studied, see: arxiv.org/pdf/2211.06120.pdf In particular, not every local system is motivic. | |
| Mar 24 at 17:10 | comment | added | JustLikeNumberTheory | @AlexanderChervov I don't think your question is analogous to "Every motivic L -function should coincide with an automorphic L - function". Asking whether every local system comes from a motive A->X is more analogous to asking whether a particular Galois representation comes from a motive, there is nothing automorphic involved. | |
| Mar 24 at 17:07 | comment | added | JustLikeNumberTheory | @AlexanderChervov Yes, it's true that one can find a Hecke eigensheaf for any local system on X, but that is not what I'm asking. I am asking if knowing this existence gives us any extra information about the original motive when the local system comes from a motive. In any case, Will Sawin's answer shows we can ask whether a motivic local system comes from a motivic eigensheaf, so this question of automorphy is not quite answered by the paper you link. | |
| Mar 24 at 10:25 | comment | added | Alexander Chervov | Probably one may ask whether ANY local system on X comes from some motive A->X ? That is opposite direction to "Every motivic L -function should coincide with an automorphic L - function" | |
| Mar 24 at 9:34 | comment | added | Alexander Chervov | Yes, it is in the abstract: "The geometric Langlands conjecture states that to each irreducible rank n local system E on X one can attach a perverse sheaf on the moduli stack of rank n bundles on X (irreducible on each connected component), which is a Hecke eigensheaf with respect to E." arxiv.org/abs/math/0012255 | |
| Mar 24 at 8:33 | comment | added | Alexander Chervov | As far as I thought there should Hecke eigensheaf on Bun(G) for ANY "local system E on X". I thought that is proved by Frenkel-Gaitsgory-Vilonen . So the first part should be true. | |
| Mar 14 at 14:58 | history | became hot network question | |||
| Mar 14 at 14:01 | answer | added | Will Sawin | timeline score: 14 | |
| Mar 14 at 1:38 | history | edited | GH from MO | CC BY-SA 4.0 |
fixed spelling of Langlands, Shimura, Frobenius
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| Mar 14 at 1:25 | history | asked | JustLikeNumberTheory | CC BY-SA 4.0 |