Timeline for Are there motives which do not, or should not, show up in the cohomology of any Shimura variety?
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22 events
| when toggle format | what | by | license | comment | |
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| Apr 6, 2020 at 10:01 | comment | added | plm | Dear Kevin and Minhyong, did you ask Laurent Clozel in the end? What is the status of conjecture CKB as of 2020? Has the coronavirus corrupted Shimura varieties enough to make it fail? | |
| Dec 9, 2019 at 16:43 | answer | added | Olivier | timeline score: 4 | |
| Sep 22, 2010 at 11:42 | comment | added | Minhyong Kim | Kevin, you are absolutely right that in my rough remarks, I ignored the 'potentiality' issue everywhere. But maybe it's sensible not to worry initially about getting the representation on the nose... I await the result of the Clozel query. I hope he's not mad at me for some sort of misrepresentation :=) | |
| Sep 22, 2010 at 9:54 | answer | added | Olivier | timeline score: 4 | |
| Sep 22, 2010 at 9:22 | comment | added | Kevin Buzzard | @Laie: I don't even know a sensible "geometric" conjecture for elliptic curves over number fields. I want them all to be associated to automorphic forms, but I don't know an analogue of the statement "if I'm an ell curve over Q then I'm covered by a modular curve" that would be expected to hold for a general number field. @David Hansen---yes! And you can find det(f) in the $H^0$ of the associated 0-dimensional Shimura variety (modulo the fact that the 0-dimensional guy doesn't quite satisfy Deligne's axioms...). But they're not enough to give you $f$, as you well know. | |
| Sep 22, 2010 at 9:19 | comment | added | Kevin Buzzard | ..for a CM elliptic curve I think I'd rather be looking for it on $GL_2$ rather than on a torus, unless, as I say, I am happy to make finite base extensions. | |
| Sep 22, 2010 at 9:19 | comment | added | Kevin Buzzard | @Minhyong: I agree with your "sound" comment! I've not heard back from Clozel yet. Regarding Mumford-Tate: consider a curve with CM over $\mathbf{Q}$. I forget conventions. If Mumford-Tate groups are defined to be connected then you're probably looking in the wrong place for your automorphic form (unless you're only "potentially" looking for it) and if they're not then there will be issues with taking "duals" because non-connectivity on the $L$ side corresponds to some sort of descent data on the other side. Maybe everything works out though! What's worrying me is that... | |
| Sep 22, 2010 at 0:03 | answer | added | Laurent F. | timeline score: 16 | |
| Sep 21, 2010 at 19:26 | comment | added | David Hansen | Given $E$ as above with associated Hilbert modular form $f$, you can find $\mathrm{Asai}(f)$ in $H^2$ of the associated Hilbert modular surface. Not quite $f$ itself, but still... | |
| Sep 21, 2010 at 19:18 | answer | added | William Stein | timeline score: 14 | |
| Sep 21, 2010 at 16:04 | comment | added | Laie | Is it possible to make a more definite statement, perhaps conjecture, about the programme outline here, not for all motives, but in the restricted case of motives that arise from Calabi-Yau varieties? | |
| Sep 21, 2010 at 15:18 | comment | added | Minhyong Kim | That could well have been the case. As a mode of proof that would render your strategy circular. But meanwhile, knowledge of such an implication would render your strategy sound. Perhaps the idea was something like this: When asserting the automorphicity of the Galois representation associated to a motive $M$, I think the Tate conjecture, Hodge conjecture, and semisimplicity conjecture should imply that the reductive group G is the dual of the Mumford-Tate group $MT$ of $M$. Perhaps there is a local system on a Shimura variety associated to $G$ or $MT$ out of which the rep. can be recovered. | |
| Sep 21, 2010 at 14:50 | comment | added | Kevin Buzzard | @Minhyong---I wonder if the standard conjectures included the one saying that the L-function of a motive should have the expected meromorphic continuation and functional equation :-) This is a pretty standard conjecture and of course it renders the plan of attack above circular. | |
| Sep 21, 2010 at 14:36 | comment | added | Minhyong Kim | Kevin: I'm reasonably certain that Clozel told me (about twenty years ago) that every motive should show up in a Shimura variety, and maybe even had an argument for deducing this from some fairly 'standard' conjectures. You could ask him and then correct me if I'm remembering incorrectly. | |
| Sep 21, 2010 at 14:30 | history | edited | Kevin Buzzard | CC BY-SA 2.5 |
added 3-line edit at bottom of q to take into account Milne's comments
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| Sep 21, 2010 at 14:27 | comment | added | Kevin Buzzard | @JS Milne: your comment is evidence that my memory is at fault and that what I was in fact told was this Albanese result, because (and I didn't let on about this in the original post) it was Blasius who was the expert mentioned in the question. I will edit the post accordingly. | |
| Sep 21, 2010 at 13:09 | comment | added | JS Milne | Blasius has pointed out that the naive generalization of the modularity conjecture fails --- there exist elliptic curves over number fields that are not quotients of the albanese of any Shimura variety --- but I don't know of any reason why the more general version (4) can't be true. (Blasius 2004 MR2058605). | |
| Sep 21, 2010 at 12:59 | comment | added | user19475 | @Kevin Buzzard: Thanks for this nice summary 1–4 of the Langlands programme! | |
| Sep 21, 2010 at 12:56 | comment | added | Kevin Buzzard | [clarification: of course in the setting above one can attach a motive because I started with $E$; I mean that in general given a level 1 eigenform I agree that it might be hard to attach a geometric object, when $F$ has even degree] | |
| Sep 21, 2010 at 12:46 | comment | added | Kevin Buzzard | I was pretty sure that one couldn't attach a motive to the level 1 eigenform. I know nothing about L-packets or U(4). I know the meromorphic continuation is known for general E/F---this is because E is potentially modular. But even proving E is potentially modular doesn't realise it in the cohomology of a Shimura variety. In fact in the situation above I assumed E was modular, so analytic continuation will be known in this setting. I wanted to emphasize that it wasn't the modularity that was the problem I was interested in, it was the Shimura variety issues. | |
| Sep 21, 2010 at 12:31 | comment | added | Olivier | The strategy of Blasius-Rogawski to construct a motive for Hilbert modular forms (base change to unitary then transfer to U(3) then to an inner form) is indeed not known to succeed in this case (because of the shape of the L-packets). I don't know if this strategy and its close cousin are known to fail (way back, Blasius and Rogawski were actually "cautiously optimistic" that it should succeed by transfer to U(4)). But you surely knew this, as you also surely know that the meromorphic continuation of the L-function of E as in your introduction is known anyway. | |
| Sep 21, 2010 at 10:52 | history | asked | Kevin Buzzard | CC BY-SA 2.5 |