One of the key steps in the 10-author paper is to prove results on local-global compatibility for Galois representations associated to torsion classes. The results proved in that paper, unfortunately, fall well-short of the optimal desired local-global compatibility statement, because there are very restrictive conditions on how the relevant primes interact with the corresponding CM field F/F^+. This is not a difficulty when it comes to modularity lifting providing one can replace F by a solvable CM extension H/F where all the required hypotheses hold. However, there are certainly other circumstances where one would like to work with a fixed F without making such a base change. One particularly interesting case is the case when the maximal totally real subfield F^+ is the rational numbers, or equivalently when F is an imaginary quadratic field. There are many reasons to be interested in this case in particular; it relates to classically studied objects (Bianchi groups) and it’s one of the very few contexts in which we have optimal results about which homology groups can have interesting torsion (in this case, you only have torsion in degree one). So how restrictive are the local-global theorems in this case? The answer is pretty restrictive — that is, they never apply directly. If one is happy to restrict to residual representations, however, then there are cheats in some cases.
Lemma: Let F be an imaginary quadratic field in which p > 3 splits, and suppose that is a congruence subgroup of of level N prime to p. Let
be a semi-simple Galois representation associated to a Hecke eigenclass in
Assume that the image of this representation contains SL_2(F_p). Then is finite flat at primes dividing p.
The point is as follows. One wants to apply Theorem 4.5.1 of the 10-author paper, but not all the conditions are satisfied. First consider the decomposed generic condition. This is guaranteed (a tedious lemma) by the big image assumption. (In fact, this hypothesis is no doubt much too strong, and possibly — in this setting where F is an imaginary quadratic field — something close to irreducibility should be enough, but I don’t really want to bother checking that now.) The more serious hypothesis in 4.5.1 is that a certain inequality holds for the degrees of various local extensions at primes dividing p in F. This inequality never holds unless there are at least three primes above p, not something that usually happens for imaginary quadratic fields. But it is possible to achieve this via a cyclic extension. For characteristic zero forms, we can appeal to cyclic base change, but this doesn’t apply for torsion classes. On the other hand, we see that we can achieve a transfer of Galois representations in the case of a cyclic extension of degree p, by the main result of this paper (I checked with at least one of the authors this preserves the property of having level prime to p). We still have to assume that p splits in F because another condition of 4.5.1 is that F contains an imaginary field in which p splits, and one can’t force this to happen after a cyclic extension H/F of (odd) degree p unless it was true to begin with. So this hypothesis will always be required if one wants to use the results of Venkatesh-Truemann in this way.
It’s an intriguing question to ask to what extent this argument could also be applied to valued representations, where is the Hecke algebra acting on mod-p classes and I is some nilpotent ideal with nilpotence of some fixed (absolute) order. This boils down to the corresponding question of how much of one sees after the cyclic degree p extension through the Venkatesh-Truemann argument. I don’t know the answer to this, but possibly a reader will. (Having done that, there are further tricks available in which one might hope to access the ring corresponding to all of rather than just the p-torsion.)