A strange continuity

Returning to matters OPAQUE, here is the following problem which may well now be approachable by known methods.

Let me phrase the conjecture in the case when the prime p = 2 and the level N = 1.

As we know from Buzzard-Kilford, in every classical weight \(\kappa\) “close enough” to the boundary of weight space, the slopes of the space of overconvergent forms are given by the arithmetic progression \(nt\) where t depends only on the 2-adic valuation of \(\kappa(5) – 1.\) Now, for each of these overconvergent forms, one obtains a Galois representation

\(\rho_{n}: G_{\mathbf{Q}} \rightarrow \mathrm{GL}_2(\overline{\mathbf{Q}}_2)\)

for every positive integer n. This gives a map from the integers \(\mathbf{N}\) considered as a discrete set to \(\mathrm{Spf}(R)\) for a deformation ring R (there is only one residual representation in this setting. Yes, it is residually reducible, but ignore this for the moment).

Problem: Show that this map from \(\mathbf{N}\) extends to a continuous map from \(\mathbf{Z}_2.\)

I’ve never done any computations in these weights, but my spidey senses says it should be true. Naturally, one should also try to work out the most precise statement where N and p are now arbitrary.

I don’t have any sense about is whether, for a fixed weight \(\kappa,\) there is actually a representation

\(\rho: G_{\mathbf{Q}} \rightarrow \mathrm{GL}_2(\mathcal{O} [[T]])\)

(for some \(\mathcal{O}\) containing enough roots of unity) whose specialization to \(T = n\) for a non-negative integer n is \(\rho_n,\) or whether the continuity is not so strong. That might be interesting to check.

More natural questions:

1. Once one has the correct formulation in fixed weight \(\kappa,\) explain what happens over the entire boundary, and at the halo.

2. I’m pretty sure that \(\rho_0\) will just be the Eisenstein series, or more accurately the Galois representation \(1 \oplus \chi,\) where \(\chi\) is determined from \(\kappa\) in some easy way involving normalizations which I don’t want to get wrong. But what is \(\rho_{-1}?\) I’m not sure if it is interesting or not. But is there any way of parametrizing this family of Galois representations so that the potentially crystalline points transparently correspond to non-negative integers?

All of this is just to say that, even for N = 1 and p = 2, there’s a lot we don’t know about the eigencurve over the boundary of weight space.