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.
For example:
Lemma: Let F be an imaginary quadratic field in which p > 3 splits, and suppose that \(\Gamma\) is a congruence subgroup of \(\mathrm{GL}_2(\mathcal{O}_F)\) of level N prime to p. Let
\(\displaystyle{\overline{\rho}: G_F \rightarrow \mathrm{GL}_2(\overline{\mathbf{F}}_p)}\)
be a semi-simple Galois representation associated to a Hecke eigenclass in \(H_1(\Gamma,\overline{\mathbf{F}}_p).\)
Assume that the image of this representation contains SL_2(F_p). Then \(\overline{\rho}\) 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 \(\mathbf{T}/I\) valued representations, where \(\mathbf{T}\) 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 \(\mathbf{T}\) 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 \(\mathbf{T}\) corresponding to all of \(H_1(\Gamma,\mathbf{Z}_p)\) rather than just the p-torsion.)
Why is the Lemma called “local global compatibility”?
Smith theory gives H_1(Gamma) a filtration (by Hecke submodules) whose graded pieces will be distributed (as Hecke subquotients) to H_1(X), H_2(X), …, H_{3p}(X), where X is an arithmetic group in SL_2(H). I don’t know what this filtration looks like in any complicated example. There may be a nontrivial constraint coming from Poincare duality. If this null answer shows that I misunderstood the question “how much of T one sees”, please tell me.
Very roughly: the Galois representation is “global” i.e. is a representation of the Galois group of a global field. One can restrict this global representation to a “local” representation, i.e. a Galois group of a local field. The expectation is that the structure of this local representation is restricted (and often determined) by the information like the level structure and the Hecke operators acting on the global representation.
For the second remark, I think even the filtration gives exactly what I ask for — you have a filtration of bounded length, and if I_m is the annihilator of the mth piece, then I = sum I_m is nilpotent with I of bounded order and T/I is then a quotient of the T at level H.
Why is I_m nilpotent? It can’t contain an idempotent of T that projects onto a complement of the piece that it annihilates?
What does the Lemma say with T/I replacing F_p-bar? Actually I still don’t know what the Lemma is about with F_p-bar there. Is rho-bar = p-torsion in an elliptic curve over F a relevant case?
Sorry, all I meant to say is that if M is a faithful T-module with a T-equivariant filtration of length N, then the corresponding action of T on gr(M) factors through T/I, where I^N = 0. I guess the point is that if I = Sum(I_m), then I contains M_m/(M_{m-1}) for all m so takes the filtered piece M_m to M_{m-1}.
The lemma says that the map rhobar: G_F —> GL_2(T/I) (for some nilpotent ideal I of fixed order depending only in p) is finite flat. Here rhobar is one of the Galois representations constructed by Scholze. If E/F is an elliptic curve which is modular of level prime to N, then it should contribute to the cohomology of GL(2)/F, and then there should be a mod-p torsion class which is the Galois representation associated to E[p], but the fact that this is finite flat is much easier to see directly.
To put it another way — one knows that this mod-p Galois representation is unramified outside p and the primes dividing the level. This is giving finer control over the ramification at p.