Local-global compatibility for imaginary quadratic fields

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.)

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Jacquet-Langlands and an old R=T conjecture

This is part 2 of a series of posts on R=T conjectures for inner forms of GL(2). (See here for part 1).

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 \mathbf{T}, 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 \mathbf{T} \oplus \mathrm{Hom}(\mathbf{T},\mathbf{Z}_p), 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 M' = \mathbf{Z}/p^2\mathbf{Z} rather than (\mathbf{Z}/p\mathbf{Z})^2.

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!

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Jacquet-Langlands and a new R=T conjecture

It is somewhat mysterious how one should formulate the Jacquet-Langlands correspondence integrally, particularly in the presence of torsion classes. Even the classical case has many subtleties including for example some results in this paper of Ribet.

In the case of imaginary quadratic fields, Akshay and I observed a number of new pathologies that don’t occur in the classical case. One of the confusing aspects was how to define a “space of newforms” which might match (in some vague sense) the cohomology of some inner form. I want to discuss here a new conjecture which is very speculative and for which I have absolutely no computational evidence. It started off as a troubling example in my mind where things seemed to go wrong in the setting of my work with Akshay, and this is the result of me trying to put down those concerns in written form. My guiding principle is that R=T in every situation, so if this doesn’t seem to work, you have to find the right definition of R (or T).

Let F be a fixed imaginary quadratic field, say of class number 1, and let P and Q be primes (of residue characteristic different from p). Suppose that

\displaystyle{H_1(\Gamma_0(P),\mathbf{Z}_p)_{\mathfrak{m}} = \mathbf{Z}_p},

where localization is done with respect to a non-Eisenstein maximal ideal of the Hecke algebra (assume all Hecke algebras are anemic for now). It can (and does) totally happen that one might have

\displaystyle{H_1(\Gamma_0(PQ),\mathbf{Z}_p)_{\mathfrak{m}} = \mathbf{Z}^2_p},

That is, at level PQ there are two old forms but nothing new either in characteristic zero or at the torsion level. In this setting, there are apparently no “newforms” of level PQ, and so one might predict that, on the quaternionic side ramified at PQ, there is no cohomology at this maximal ideal. This is certainly true in characteristic zero by classical Jacquet-Langlands. But it is false integrally! In particular, suppose that the corresponding residual representation

\displaystyle{\overline{\rho}: G_F \rightarrow \mathrm{GL}_2(\mathbf{F}_p)}

has the property that the image of Frobenius at Q has eigenvalues with ratio N(Q). Then one indeed expects a contribution on the non-split side. Akshay and I managed to find an interpretation of this result by giving a “better” definition of the space of newforms as the cokernel of a transfer map:

\Phi^{\vee}: \displaystyle{H_1(\Gamma_0(P),\mathbf{Z}_p)^2_{\mathfrak{m}} \rightarrow H_1(\Gamma_0(PQ),\mathbf{Z}_p)_{\mathfrak{m}}},

and this can have interesting torsion even in the context above. In fact, by a version of Ihara’s Lemma, one can (and we did) compute that the order of the cokernel in this case will be exactly the order of

\mathbf{Z}_p/(a^2_Q - 1 - N(Q)) \mathbf{Z}_p,

and (again in this precise setting) Akshay and I predicted that this should have the same order as the corresponding localization at the same maximal ideal on the non-split side. (In the Eisenstein case, this is not true, and one sees contributions from various K_2 groups). We even prove a few theorems which prove results of this form taking a product over all maximal ideals of the Hecke algebra.

But even in this example, something a little strange can happen. In particular, I want to argue in this post that there are two natural definitions of the appropriate global deformation ring, and in order to have a consistent theory, one should consider both of them. To remind ourselves, we now have two modules, one, defined in terms of the cokernel above, call it M, and then the cohomology localized at the appropriate maximal ideal on the non-split side, which we call M’.

What should we predict about M? The first prediction is that the image of the Hecke algebra should be precisely the universal deformation ring R_Q which records deformations that are Steinberg at Q (and what they should be at the other places). But what does Steinberg at Q even mean for torsion representations? There are basically two types of guesses for the corresponding local deformation ring, and correspondingly two guesses for the associated global deformation ring.

  1. A deformation ring defined in terms of characteristic polynomials. In particular, the maximal quotient of R_Q which corresponds to classes unramified at Q is the unramified deformation ring where the characteristic polynomial of Frob_Q is (X-1)(X-N(Q)).
  2. A more restrictive ring in which (on this same unramified quotient) the image of Frob_Q must actually fix a line.

These certainly will have the same points in characteristic zero, but they need not a priori coincide integrally. And this will save us below.

Returning to the corresponding global deformation rings (which should be framed, but now ignore the framing), call the corresponding rings R_Q and R’_Q. There is a surjection from R_Q to R’_Q.

Now we make the following conjecture on the smell of an oily rag:

Conjecture: The Hecke action on M has image R_Q while the Hecke action on M’ has image R’_Q.

I base this conjecture entirely on the following thought experiment.

Let’s suppose for convenience that N(Q) is not -1 mod p. This implies that a_Q is congruent to precisely one of 1+N(Q) or its negative — assume the former. Then the “space of newforms” M as we define it (under all the hypotheses above) will be actually be isomorphic to

\mathbf{Z}_p/(a^2_Q - 1 - N(Q)) \mathbf{Z}_p =:\mathbf{Z}_p/\eta \mathbf{Z}_p,

because one of the factors will be a direct summand. (The case when N(Q) = -1 mod p is no problem but one has to break things up more using the Hecke operator at U_Q which I am ignoring.) So the claim in this case is that R_Q is isomorphic to this ring. What about R’_Q? Let us consider two possibilities.

(added: Note that if N(Q) =/= 1 mod p then R_Q=R’_Q, so we are assuming that N(Q)=1 mod p in the examples below.)

Example 1: Suppose that a_Q – 1 – N(Q) is exactly divisible by p^2, and that

\rho(\mathrm{Frob}_Q) = \left(\begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}\right) \mod p.

In this case, the non-split property implies that the corresponding matrix modulo p^2 will always have 1 as an eigenvalue, so the prediction is that R_Q = R’_Q.

Example 2: Suppose that a_Q – 1 – N(Q) is exactly divisible by p^2, and that

\rho(\mathrm{Frob}_Q) = \left(\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}\right) \mod p.

In this case, the split condition and the assumption that a_Q – 1 – N(Q) is exactly divisible by p^2 force the lift to be of the form

\rho(Frob_Q) = \left(\begin{matrix} 1 + a p & p b \\ 0 & 1 + c p \end{matrix}\right) \mod p^2.

where a and c are non-zero. In particular, 1 will never be an eigenvalue. So in this case, one predicts that R_Q = Z/p^2Z but R’_Q = Z/pZ.

So how do we see this in terms of R=T and Jacquet-Langlands and our Conjecture above? First of all, my paper with Akshay suggests indeed that |M’|=|M|= p^2, and certainly M’ should be an R_Q-module. But now the following should happen:

  1. In Example 1, we should have multiplicity one, and so M’ should be free of rank 1 over R_Q = R’_Q.
  2. In Example 2, we should have multiplicity two, following Ribet (Helm, Cheng, Manning…), since multiplicities should be determined by local conditions, and in particular multiplicities should arise exactly when primes which ramify in the quaternion algebra are split and such that the image of the corresponding Frobenius is scalar. Hence M’ should be free of rank 2 over R’_Q in this case.

In particular, the Hecke action on M’ should factor through R’_Q in both cases, and R_Q does not act faithfully. Perhaps this conjecture is worth a computation!

Update: Read Part 2 of this series.

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And then there were 8:

News from Minnesota on the 8-author paper has arrived!

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This is the current state of my inbox:


That either makes me very efficient and up to date, or simply very unpopular (or both!)

I’m ready for the new year!

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New Results In Modularity, Christmas Update

It’s a Christmas miracle! Keen watchers of this blog will be happy to learn that the 10 author paper discussed here and here is now available. (And just in case you also missed it, you can also find the other modularity result discussed here at this link.)

You may ask — what took so long? Well, it’s quite long, as you might notice. Also, as it turns out, one version of this paper was almost ready quite some time ago, but then quite a bit of time was spent improving the ordinary local-global compatibility statements and thus the ordinary modularity lifting theorems so that they would apply to small p. The hypothesis in the Fontaine–Laffaille chapter is that p > n^2, which is not ideal if one wants to take n = 2 and p = 2 or 3 (which indeed one might have occasion to do), but now the hypothesis in the ordinary section is much less restrictive, so yay for that.

The paper relies on the (as yet unavailable) work of Caraiani-Scholze; I hope to be able to have a post explaining a little bit more about that paper relatively soon.

Finally, I wanted to make one remark about credit. I think with an almost 200 page paper (*) that proves a number of quite nice results, there’s enough credit to go around among 10 different authors. On the other hand, I think it’s worth pointing out (perhaps even incumbent upon me to say) that the younger authors of this paper did more than their fair share of work — both creatively, intellectually, and practically.

I hope you enjoy reading this paper as you tuck into your plum pudding, unless you happen to be one of those peculiar people who doesn’t much care for the thought of a pudding consisting of dried fruits held together by eggs, breadcrumbs, cognac, and a bunch of fat from beef kidneys, in which case your soul must be even more desperately in need of nourishment and thus you will be even more in want of a paper such as this.

(*) to be fair, almost 200 pages using a LaTeX style file with generously wide margins — it’s probably only more like 150-160 in the standard format.

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Guess That Tune!

I have recent upgraded my wheels from “classic” (20th century corolla, slightly used) to “contemporary” (21st century corolla, also slightly used). Possibly the last straw was when I found the article Toyota Recalls 1993 Camry Due To Fact That Owners Really Should Have Bought Something New By Now, and then noticed that the article was already more than three years old. Perhaps the biggest difference is that the car is not in imminent danger of catastrophic collapse. But a close second is that I can now play music through my iPhone as I drive to work. Currently they are on random shuffle, which means that I get to play “guess that tune” for my current collection (5000 songs or so). The rules of the game are a little vague. How close does the answer need to be? Suppose it’s a Bach French suite. Do I have to name the suite? Do I have to name the pianist? Do I have to remember if the movement is an bourrée or a gigue? Schubert Lieder — Do I have to distinguish my Schwanengesangs from my Die schöne Müllerins? Do I have to know the key of a Prelude and Fugue (Bach or Shostakovich)? The number of a Goldberg variation? (and whether it is Gould in ’55 or Gould in ’81).

I don’t think I have complete answers to these questions; some of it depends a little bit on how familiar I am with the actual music. For example, “ah, that must be one of those crappy Vivaldi flute concertos played by Rampal that I downloaded from Jacob once” is a perfectly satisfactory answer. So is “The Gigue from the second English suite — and it must be Perahia, because he plays this one phrase with a definite whoops-sie-doo lilt.”

Some observations so far. I’m barely scoring above random on St Matthew passion versus the St John Passion (especially on random 25 second recitatives). It’s not easy to distinguish early Beethoven (piano) from late Mozart. This is especially true when your Beethoven collection is complete enough to have all of the crappy Beethoven. I do much better on piano music than chamber music. Sometimes it’s much easier to guess after listening for 30 seconds rather than 5 seconds. Unfortunately, I am not technically competent enough to cut out snippets from audio files in order to let you play this game, but, then again, you will probably have more fun anyway doing it on your own music…

But I can at least post the following links, and you can see whether you agree with me on the whoops-sie-doos (5 seconds in to the recording, or, if the link takes you to the beginning, at 1:35 and 1:40 respectively)

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