This is part 2 of a series of posts on R=T conjectures for inner forms of GL(2). (See here for part 1).
(Edit: this is still incorrect and there should have been a part 3, but I’ve been distracted… in conversations with Boxer, Emerton, and Gee shortly after this post, all issues were resolved. Jeff Manning also independently found the correct formulation.)
I feel that I should preface this post with the following psychological remark. Occasionally you have the germ of an idea at the back of you mind that you sense is in conflict with your world view. Perhaps you try subconsciously to banish it from your mind, or perhaps you are drawn towards it. But inevitably, the idea breaks through your consciousness and demands to be addressed. The game is now winner-takes-all — either you can defeat the challenge to your world view, or you will be swallowed up by this new idea an emerge a new person. This is how I came face to face with the non-trivial multiplicities in cohomology for non-split forms of GL(2) over an imaginary quadratic field. Part of me somehow, unconsciously, worried about the conflict between extra multiplicities on the one hand and, on the other hand, the “numerical” equality between the space of “newforms” on the split side with the corresponding space on the non-split side (this equality is not known for each maximal ideal of the Hecke algebra, but rather the “averaged” version over all maximal ideals is the topic of my paper with Akshay). Then, earlier this week, I turned my face directly towards the problem and admitted its existence, which lead to the previous post. But now… there may be a way to defeat the beast after all!
Here is the issue. I talked last time about two types of local framed Steinberg deformation rings at l=1 mod p. The first was defined by imposing conditions on characteristic polynomials, but the second was a more restrictive quotient which demanded the existence of an eigenvalue which was genuinely equal to 1. This modification seemed to pass some consistency checks, and more importantly resolved the compatibility issue between having both the equality |M| = |M’| but also having M be cyclic whilst M’ was not. Then I went away for a few days and was distracted by other math, until I flew back to Chicago this evening. While on the plane, I tried to flesh out the argument a little more by writing down more carefully what these two deformation rings R (and its smaller quotient R’) were like. And here’s the problem. It started to seem as though this quotient R’ didn’t really exist — after all, demanding the existence of an eigenvector without pinning it down in the residual representation is a dangerous business, and runs into exactly the same issues one sees when trying to give an integral definition of the ordinary deformation ring for l=p. Then I thought a little more about the ring R, and it turns out that, for all the natural integral framed deformation rings one writes down, the ring R is a Cohen-Macaulay normal integral domain! In particular, since R’ has to be of the same dimension of R, this pretty much forces R to equal R’. So it seems that my last post is completely bogus.
So what then is going on? When you have eliminated the impossible, whatever remains, however improbable, must be the truth. It is impossible that R does not equal so I can only conclude the improbable — that even when the representation rhobar is unramified at l and the image of Frobenius at l under rhobar is scalar, the multiplicity on the quaternionic side ramified at l will still have multiplicity one. In other words, the local multiplicity behavior will be sensitive to the archimedean places. This is not what I would (or did) guess, but I cannot see another way around it. So, at the very least, we should investigate this assumption more closely.
Let’s talk about two situations where multiplicity two occurs. The first is in the Jacobian J_1(Np) for mod-p representations which are ramified at p. In this case, the source of multiplicities is coming from the fact that the local deformation ring R is Cohen-Macaulay but not Gorenstein. On the other hand, the stucture of the Tate module is well understood to be of the form and so the multiplicity can (ultimately) be read off from the dualizing module of R. This is what happens in my paper with David Geraghty. The second, which is something I should have paid more attention to last time, is in the work of Jeff Manning (I can’t find a working link to either the paper or to Jeff!). The setting of Manning’s work is precisely as above: one has l=1 mod p and one is looking at the cohomology of an inner form of GL(2)/F. The only difference is that F is totally real and the geometric object is a Shimura curve. The corresponding local deformation ring R — which is basically the corresponding ring R above — is Cohen-Macaulay but not Gorenstein. On the other hand, one doesn’t now know what the structure of the Jacobian is as a module over the Hecke ring. Manning’s idea is to exploit the fact that, in his setting, the module M is reflexive (and generically of rank one), and then by studying the class group of R, pin down M exactly. But here is the thing. The reflexivity of M is coming, ultimately, from the fact that the cohomology group H^1 for Shimura curves is self-dual. And this is fundamentally not the case for these inner forms for GL(2) over an imaginary quadratic field, where the cohomology is spread between H^1 and H^2. So this is where the archimedean information can change the structure. At this point, I am pretty much obligated to make the following conjecture.
Conjecture: For inner forms of GL(2) over an imaginary quadratic field, and for a minimal rhobar which is irreducible and finite flat at primes dividing p > 2, the multiplicity of rhobar in cohomology is one. Moreover, the correpsonding module M’ of this cohomology group localized at this maximal ideal is isomorphic (as R-modules and so as Z_p-modules) to the space of newforms on the split side, as defined in the last post.
To put it another way, in Example 2 of the previous post, I am now forced to say that rather than
To reiterate from last time — perhaps this conjecture is worth a computation!
I guess we shall have to wait a few days to see whether there will be a part 3!