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A Sunday question for specialists of eigenverieties:

In their important paper "the eigencurve", Coleman and Mazur globalized the earlier construction of Coleman's families, constructing a beautiful eponymous rigid analytic space that parametrizes all systems of Hecke eigenvalues of finite-slope overconvergent modular forms. Since them many generalizations of this construction have been performed (the "eigenvarieties") which each time appear as globalizations of local constructions generalizing Coleman's families. This process of globalization has even been axiomatized by Buzzard ("the eigenvariety machine").

Yet, I wonder:

What are the benefits of working with a global object (which is considerably more difficult to construct and to deal with) rather than just the local objects with which it is constructed (the families of Coleman and their generalizations) ?

Of course, having a global, canonical, object is much more satisfying on esthetic grounds. As a mathematician formed after Grothendieck's revolution, this reason alone would be for me a sufficient one to consent the effort of constructing global eigenvarieties. But my question is meant to be understood a little bit more specifically:

What are the applications, or expected applications (to our knowledge of the arithmetic of automorphic forms, Galois representations, L-functions, etc.) of the global existence and geometry of eigenverities that are not already consequences of the existence and geometry of their local pieces?

Of course, there are already an enormous amount, still growing fast, of arithmetic informations obtained from the local pieces of eigenvarieties. But what for the global structure? Let me mention the only one I know: the global existence of the eigencurve (say) is necessary to be able to attached to any overconvergent finite slope modular form a Galois representation. With Coleman's families alone, we could construct those representations only for these forms having a weight sufficiently close p-adically to a non-negative integer (for example the one with negative weights). Yet I find this application not very convincing, as why do we care about overconvergent form with weight far away from integers except for their being the "flesh" of the eigencurve?

So what other applications do you have in mind?

(edited for one typo)

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When I told Coleman what I knew about the $p=2$ $N=1$ eigencurve (that it was "proper" over weight space and looked like a disjoint union of annuli near the boundary) he said "great, so now you can glue a disk to each of the annuli and get something proper over projective 1-space, and that might tell you something". I never worked out what he was thinking though. I think he might have been hoping to say something about components. But I agree -- you have asked a nice question. –  Kevin Buzzard Sep 11 '11 at 20:00
Maybe I'm being naive, but aren't the various Galois-theoretic characterizations of reps coming from overconvergent forms one reason to care about all overconvergent forms? –  Kevin Ventullo Sep 11 '11 at 20:25
I don't think anyone has any idea on how to characterize Galois representations coming from <i> all </i> overconvergent eigenforms. –  Michael Sep 11 '11 at 22:04
However, without the global construction of the eigencurve, the local version of the same proof would lead to the same result restricted to, say, rep. whose difference of Sen weights is in $Z_p$. (I am sure it is so for Kisin's part of the proof. I don't know well enough yet Emerton's part, but I would be surprised if it was not "local" -- and that would be a great answer to my question). Of course this is less elegant, and elegance is important. But, from a certain point of view, the other Galois representations are anecdotical. (to be continued) –  Joël Sep 12 '11 at 1:29
This point of view (that I don't share completely) is the one for which what really matters is the classical modular forms or the motivic Galois representations, and that the overconvergent modular forms and the general Galois representation are but a tool to study the former. From this point of view, what is great in Emerton's result is the Fontaine-Mazur conjecture, and if it can be proved (as I believe, but tell me if I am wrong) by considering only Coleman's families instead of the eigencurve, then the whole thing is not an essential application of global eigenvariety and an answer. –  Joël Sep 12 '11 at 1:38
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2 Answers

up vote 6 down vote accepted

This is something I've told people privately for a while, and now have enough ingredients written up (jointly with Liang Xiao) to claim on the web somewhere: the validity of the parity conjecture is constant in $p$-adic analytic families. (Fine print: the family must be symplectic self-dual, and be equipped with a sort of Panchishkin triangulation, but that's all.) Now that the triangulation of the entire eigencurve has been constructed, and the conjecture is known in weight two by any number of authors, it follows that the parity conjecture holds for any finite-slope form lying on an irreducible component of the eigencurve that admits a classical weight two point. Thus, the parity conjecture for all finite-slope forms reduces to the claim that Buzzard's observation noted above in $p=2,N=1$ holds generally (apply Coleman's classicality theorem to the low-slope, weight two points near the boundary), which is a question of the global geometry of the eigencurve.

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Jay, nice to see you there. That's a very convincing answer. –  Joël Oct 24 '13 at 22:30
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On p. 4 of Kisin's paper "Overconvergent modular forms and the Fontaine-Mazur conjecture", he explains the possibility of proving modularity lifting theorems via "analytic continuation along the eigencurve". This seems to require a global point of view, since it's predicated on understanding (at the very least!) the connected components of the global eigencurve.

Also, Emerton's completed cohomology is a very global object, in the sense you're asking for: applying his locally analytic Jacquet module functor to the locally analytic vectors in $\widehat{H}^1$ gives the whole (reduced) eigencurve for $\mathrm{GL}_2/\mathbf{Q}$, no gluing required! (I hope Professor Emerton will correct any misrepresentations I have made of his work, if he reads this.)

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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. –  Joël Sep 12 '11 at 19:16
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... –  David Hansen Sep 12 '11 at 20:17
... 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. –  David Hansen Sep 12 '11 at 20:21
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 ... –  Emerton Sep 12 '11 at 20:30
... 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 –  Emerton Sep 12 '11 at 20:32
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