Base Change for Eigenvarieties Let $E/F$ be a Galois extension of number fields, and $G$ a reductive group over $F$. If Langlands Base Change is known for $G/F$ and $G/E$, and moreover the eigenvarieties for $G/F$ and $G/E$ have been constructed, is there a rigid map between the eigenvarieties which interpolates the base change?
Assuming the answer to the previous question is yes, will the map be a closed immersion? What is known about the image with respect to the subvariety of Gal($E/F$)-invariants?
Finally, are there any instances where we have such a map between eigenvarieties but base change is not known?
Sorry for the barrage of questions; I would be happy with even a partial answer to the first.
 A: This is all a bit complicated -- the theory is still in its infancy and some arguments aren't quite as smooth as they should be.
If all you know is that "the eigenvarieties have been constructed" then you're in a hopeless situation -- in fact in some sense I don't even know what this statement means. You need to know some facts about the eigenvarieties before you can prove anything. Let's say for example that you're in the happy situation where classical points are dense (for example, $G$ compact mod centre, or perhaps you've defined an eigenvariety to have that property by taking the closure of the classical points in an a priori bigger object). Then you might be able to get somewhere (see next para). However there are examples of Hida families over non-totally real bases where there appear to be natural $p$-adic objects where classical points do not seem to be dense, so then things might be really tough.
If classical points are dense, then there is still another big problem: as far as I know Langlands Base Change (and functoriality in general) only transfers packets to packets. However, eigenvarieties don't parametrise packets, they parametrise systems of Hecke eigenvalues. If you are dealing with inner forms of $GL(2)$ then packets have size 1 so Chenevier doesn't see this problem and he can prove the best possible theorem (this is one of the papers you cite). If however there are packets then one runs into combinatorial issues -- Flicker finds himself with such problems in the other paper you cite. So, even if classical points are dense, things might not be so easy.
I have a student, Judith Ludwig, who is making some headway with issues of this nature. The questions can be quite delicate.
