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 $\ell=p$?



What is the nature of the locus in the eigencurve associated to some conditions on the associated automorphic representation (at $p$)?
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?


