Timeline for Global applications of eigenvarieties
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
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| Sep 12, 2011 at 22:35 | vote | accept | Joël | ||
| Oct 24, 2013 at 20:58 | |||||
| Sep 12, 2011 at 20:34 | comment | added | Emerton | Dear David, The eigencurve and locally analytic Jacquet modules appear in section 7.2 of local-global compatibility, "Applications to a conjecture of Kisin". This is the argument I am referring to in my previous comment. Regards, Matt | |
| Sep 12, 2011 at 20:32 | comment | added | Emerton | ... eigensurface (i.e. eigencurve together with wild twists), and in particular, the fact that this space is equidimensional of dimension two (with one dimension being given just by twisting). This seems to be a global application. Best wishes, Matt | |
| Sep 12, 2011 at 20:30 | comment | added | Emerton | Dear David and Joel, If one just wants to study Fontaine--Mazur (or even local-global compatibility for $p$-adically completed $H^1$), I don't think that you need the eigencurve; some form of the infinite fern (which works just with Coleman families) will be enough. On the other hand, in the proof of Kisin's conjecture one wants to show that the support of the locally analytic Jacquet module of $\widehat{H}^1$ is precisely the set of twists of finite slope o.c. eigenforms. For this, one uses (along with other ingredients) the fact that these forms are parameterized by the points of the ... | |
| Sep 12, 2011 at 20:23 | comment | added | David Hansen | (In case it's not clear, the main body of my answer was referring to Emerton's 2006 Inventiones paper on eigenvarieties, and not the local-global compatibility paper.) | |
| Sep 12, 2011 at 20:21 | comment | added | David Hansen | ... but my interpretation is that Emerton uses a "local at $p$ version of Kisin's program", namely a suitably strong $p$-adic local Langlands correspondence and the understanding of trianguline representations therein ("the local eigencurve"), as the key tool in analyzing $M(\rho)$ even though the latter is defined in terms of a single characteristic zero Galois representation. | |
| Sep 12, 2011 at 20:17 | comment | added | David Hansen |
Joel, I am quite sure of what I have written, but I am not sure about the relation between Emerton's local-global compatibility paper and the eigencurve. Emerton avoids the eigencurve: rather he basically shows that $M(\rho)=\mathrm{Hom}_{E[G_{\mathbf{Q}}]}(\rho,\widehat{H}^1)$, as a representation of $GL_2(\mathbf{Q}_p)$, has locally algebraic vectors iff $\rho$ is de Rham up to twist, and this basically forces $M(\rho)$ to contain images of $H^1$'s at finite level, so also classical modular forms by Eichler-Shimura. I'm not sure if the eigencurve lurks here somewhere...
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| Sep 12, 2011 at 19:16 | comment | added | Joël | Thanks, David. I have been thinking since four hours to what you say. Basically, you're suggesting that Emerton's methods are essentially global in nature (this is what you say literally in your 2nd paragraph, but even the 1st is about Emerton, since it is he who eventually realized this program laid out by Kisin in the reference you quote, no? - in Kisin's paper, things are essentially local). If this is so, that's a perfect answer to my question. But I'd like to be sure (since you don't seem so sure yourself) and to understand why. Anyone? Also, my careful reading of E's papers is overdue. | |
| Sep 12, 2011 at 14:38 | history | answered | David Hansen | CC BY-SA 3.0 |