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I've chatted informally with some folks about this question before and gotten some very nice insights, but I thought I'd toss it out to a wider audience because it is a continuing curiosity of mine.

Roughly, here's what I have in mind: Let $\mathcal{E}$ denote the eigencurve of some tame level $N$. At classical points, one has attached to the associated form an automorphic representation $\pi = \otimes \pi_\ell$. For $\ell\neq p$, there are results (though I've forgotten the author(s) and couldn't dig up the reference in the course of posting this) about the loci in the eigencurve corresponding to various conditions on the $\pi_\ell$. By conditions here I'm rather open-minded - things such as special, supercuspidal, bounds on conductor, etc., are all of interest to me.

My question is: What is known about the analogous questions for $\ell=p$?

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    $\begingroup$ Could you be a little more specific about the type of condition you are looking for? For all the conditions I can think of, I imagine you would already know the answer, e.g. (locally reducible => ordinary locus), (potentially semi-stable => classical points [by a thm of Kisin]), (trianguline => the entire curve, etc.) $\endgroup$
    – Michael
    Commented Sep 3, 2011 at 4:55
  • $\begingroup$ @Michael: I was thinking of conditions on the automorphic side of things, like the three I mentioned. Perhaps these are only defined at classical points to begin with, but I want to know how to think about the locus where these point naturally lie. Something along the lines of "these are exactly the classical points in a certain union of irreducible components," though I don't think that the answer for the cases I mentioned. $\endgroup$
    – Ramsey
    Commented Sep 3, 2011 at 16:11
  • $\begingroup$ If you are looking on the automorphic side, you presumably want to vary $p$-adic $\mathrm{GL}_2(\mathbf{Q}_p)$-representations. My understanding is that $p$-adic Langlands programme shows that one wants to attach a $p$-adic Banach space $B(\rho_p)$ to the local Galois representations $\rho_p$ that will vary nicely along the eigencurve, although $B(\pi_p)$ is a little more complicated than its classical avatar $\pi_p$ (which contains less information, in general). I would recommend looking at Breuil's ICM address if you haven't already done so. $\endgroup$
    – Michael
    Commented Sep 3, 2011 at 19:46
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    $\begingroup$ Dear Nick, Basically, the $\pi_p$ are all principal series. This is the representation-theoretic interpretation of $U_p \neq 0$. More precisely, $U_p \neq 0$ means (a) unramified twist of Steinberg; or (b) principal series with at least one of the inducing characters being unramified. Personally, I find the eigensurface (all wild twists of the eigencurve) more natural: then the $\pi_p$ at classical points are precisely principal series or twists of Steinberg. Best wishes, Matt $\endgroup$
    – Emerton
    Commented Sep 4, 2011 at 4:27
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    $\begingroup$ Dear Ramsey, the special points are certainly discrete, because the map $\mathcal{E} \rightarrow \mathbb{C}_p$ given by $f \mapsto (a_p)^{-1}$ is continuous, and the special points map to the discrete set $\pm p^n$ for $n \in \mathbf{N}$. The question "does an irreducible component contain a special point" is a little tricky. One reason is that it is hard to know what the components are (are there finitely many or infinitely many?). $\endgroup$
    – Michael
    Commented Sep 4, 2011 at 17:49

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Since the eigencurve only sees classical modular forms with non-zero $U_p$-eigenvalue, there are strong restrictions on the local factor at $p$ for the attached automorphic representations (e.g. you won't ever see supercuspidals). I think the situation is that if you fix a Nebentypus, with conductor $p^r$ at $p$, then the automorphic representations coming from classical eigenforms with non-zero $U_p$ eigenvalue must have local factor at $p$ either a) Principal series with conductor $p^r$ or b) Special of conductor $p$ (so $r=0$ in this case). (One can find these sorts of calculations in Casselman's article "On representations of $GL_2$ and the arithmetic of modular curves." in one of the Antwerp volumes.)

Also, any classical point sufficiently close to (but not equal to!) one of the $p$-special classical points will actually be unramified principal series (by local constancy of the slope, since the slope of a weight $k$, $p$-special point has to be $(k-2)/2$), so every component will contain principal series points.

As a complement to Michael's comment, bearing in mind Emerton's approach to constructing the eigencurve and his results on local-global compatibility in the $p$-adic Langlands programme, it might be natural to think about certain locally analytic representations of $GL_2(\mathbb{Q}_p)$ varying over the eigencurve - e.g. for the two $p$-stabilised $U_p$-eigenforms coming from a classical eigenform with level prime to $p$, one has two different locally analytic principal series representations over the two associated points on the eigencurve. One can also see the locally analytic Jacquet modules (as defined by Emerton) of these representations varying over the eigencurve.

PS Maybe you're thinking of Alex Paulin's (http://math.berkeley.edu/~apaulin/) thesis for the $l \ne p$ case?

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  • $\begingroup$ Yeah, I probably shouldn't have tossed "supercuspidal" into my "open-minded" list of properties in this case. A little too open-minded... At any rate, these are great comments. Thanks! $\endgroup$
    – Ramsey
    Commented Sep 4, 2011 at 12:47
  • $\begingroup$ I think you forgot to mention the work of James Newton on level raising at $l \ne p$. $\endgroup$
    – Michael
    Commented Sep 4, 2011 at 17:55
  • $\begingroup$ Dear Michael, yeah I guess I should have mentioned my own work too! $\endgroup$
    – jnewton
    Commented Sep 6, 2011 at 13:59

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