Galois representations along eigenvarieties

Question

This question is about the status of the following.

Meta-hypothesis. Let $X$ be an irreducible component of an eigenvariety. Then there exist: (a) a pseudo-representation/character $\psi$ along $X$, specializing to what it should at classical points; (b) a "cover" $f\colon X' \rightarrow X$, a locally free sheaf $V$ on $X'$, and an ${\scr O}_{X'}$-linear Galois action on $V$, such that that $f^*\psi$ is the pseudocharacter associated with $V$.

Questions:

1. What instances of this statement are known, and with what constructions?
2. In what generality do we expect the hypothesis to hold, and under what meaning of "cover" (and "eigenvariety")?
3. What about the Coleman-Mazur-Buzzard eigencurve of tame level $N$? Can one take $X'$ to be the normalization of $X$, and if so, are the details written anywhere?

Some remarks. There are quite general results on (a) by Chenevier, Bellaiche-Chenevier,...; the less clear part (to me) is (b). Two constructions which come to mind and work "on the nose" ($X'=X$) in special cases are the following: (1) for the eigencurve of level 1, take (the Jacquet module of) completed cohomology; (2) if the mod $p$ reduction $\overline{\psi}$ of the universal $\psi$ on $X$ corresponds to an irreducible mod $p$ Galois representation $\overline{\rho}$, then the generic fibre of the Spf of the universal deformation ring $T_{\overline{\psi}}=T_{\overline{\rho}}$ admits a map from $X$ and carries a universal Galois-sheaf $W$, which can then be pulled back to a Galois-sheaf $V$ on $X$.

Answer 1

The answer depends what kind of "cover" you need down the road. For a strong definition of cover, like a "Zariski cover", the answer is no in general. I believe it is still no for an "étale cover" or "fpqc cover".

Now if you ready to consider as covers not only Zariski covers but any proper and birational map $X' \rightarrow X$, then the answer is yes in full generality, and for general reasons which have not much to do with eigenvarieties. See Lemma 3.4.2 in my book with Chenevier (Astérisque 324) and how it is used in the proof of Theorem 3.4.1. (I can give more details if needed).

Answer 2

Are you familiar with the paper "Overconvergent Eichler--Shimura isomorphisms" by Andreatta--Iovita--Stevens? In section 3 of this paper they give a variant of the "modular symbol" construction of the Coleman--Mazur eigencurve, which naturally gives rise to a coherent sheaf of Galois representations. They only give the argument locally over small affinoid patches in weight space; but Shanwen Wang and (independently) David Hansen have checked that the argument globalises to the cuspidal eigencurve.

In more general settings, you have to decide two things: firstly, what kind of eigenvariety machine you want to use; secondly, what kind of Galois representations you want to see. For instance, if you throw Emerton's machine at $GL_2$ over a totally real field, you'll get an eigenvariety with a coherent sheaf of Galois representations, and (assuming the degeneration of Emerton's spectral sequence) this will interpolate the Galois representations appearing in the etale cohomology of Hilbert modular varieties. But these aren't the "standard" Galois representations of Hilbert eigenforms; they're the tensor inductions of these to $G_\mathbf{Q}$. If you use a quaternion algebra split at all finite places and ramified at all but one infinite place, you'll get essentially the same eigenvariety, but with a different sheaf of Galois representations -- this will give you the standard 2-dimensional reps.