## Jean-Marc Fontaine, 1944-2019

The results which generate the most buzz in mathematics are usually those which can be expressed in an elementary (or at least pithy) way to a general mathematical audience. It is certainly true that such results may be profound (see Wiles, Andrew), but this is not always the case. An indirect consequence of this phenomenon is that there are mathematicians who are considered absolute titans of their own field, but who are less well-known by the broader mathematical community. Fontaine, who died this year, might be considered one of these people. Fontaine will forever be associated with p-adic Hodge theory, a subject which is absolutely central to algebraic number theory today. While the initial seed of this subject came from Tate’s paper on p-divisible groups, a huge part of its development was due to Fontaine over a period of 30 years (both in his solo papers and in his joint work). The usual audience for my posts is experts, but on the rare chance that someone who knows less p-adic Hodge theory than me reads this post, let me give the briefest hint of an introduction to the subject.

For a smooth manifold M, de Rham’s Theorem gives an isomorphism

$H^n_{\mathrm{dR}}(M) \rightarrow H^n(M,\mathbf{R}) = H_n(M,\mathbf{R})^{\vee}$

which can more naturally be phrased as that the natural pairing between (classes of) closed forms $[\omega]$ and (classes of) paths $[\gamma]$ given by

$\displaystyle{\langle [\omega],[\gamma] \rangle = \int_{\gamma} \omega}$

induces a perfect pairing on the corresponding (co-)homology groups. The class of paths in homology has a very natural integral basis coming from the paths themselves. For a general M, the de Rham cohomology has no such basis. On the other hand, if M is (say) the complex points of an algebraic variety over the rational numbers, then there are more algebraic ways to normalize the various flavours of differential forms. To take an example which doesn’t quite fit into the world of compact manifolds, take X to be the projective line minus two points, so M is the complex plane minus the origin. There is a particularly nice closed form $dz/z$ on this space which generates the holomorphic differentials. But now if one pairs the rational mutiples of this class with the rational multiples of the loop $\gamma$ around zero, the pairing does not land in the rational numbers, since

$\displaystyle{\int_{\gamma} \frac{dz}{z} = 2 \pi i}.$

In particular, to compare de Rham cohomology over the rationals with the usual Betti cohomology over the rationals, one first has to tensor with a bigger ring such as $\mathbf{C},$ or at least with a ring big enough to see all the integrals which arise in this form. Such integrals are usually called periods, so in order to have a comparison theorem between de Rham cohomology and Betti cohomology over $\mathbf{Q},$ one first has to tensor with a ring of periods.

It is too simplistic to say that p-adic Hodge theory (at least rationally) is a p-adic version of this story, but that is not the worst cartoon picture to keep in your mind. Returning to the example above, note that the period is a purely imaginary number. This is a reflection of the fact that some arithmetic information is still retained, namely, an action of complex conjugation on the complex points of a variety over the rationals is compatible (with a suitable twist) with the de Rham pairing. A fundamental point is that, in the local story, something similar occurs where now the group $\mathrm{Gal}(\mathbf{C}/\mathbf{R}) = \mathbf{Z}/2 \mathbf{Z}$ generated by complex conjugation is replaced by the much bigger and more interesting group $\mathrm{Gal}(\overline{\mathbf{Q}}_p/\mathbf{Q}_p).$ Very (very) loosely, this is related to the fact that p-adic analysis behaves much better with respect to the Galois group, for example, the conjugate of an infinite (convergent) sum of p-adic numbers is the sum of the conjugates. In particular, there is a Galois action on the ring of all p-adic periods. So now there is a much richer group of symmetries acting on the entire picture. Moreover, the structure of the p-adic differentials can be related to how the variety X looks like when reduced modulo-p, because smoothness in algebraic geometry can naturally be interpreted in terms of differential forms.

So now if one wants to make a p-adic comparison conjecture between (algebraic) de Rham cohomology on the one side, and etale cohomology (the algebraic version of Betti cohomology) on the other side, one (optimally) wants the comparison theorem to respect (as much as possible) all the extra structures that exist in the p-adic world, in particular, the action of the local Galois group on etale cohomology, and the algebraic structures which exist on de Rham cohomology (the Hodge filtration and a Frobenius operator), and secondly, involve tensoring with a ring of periods B which is “as small as possible”.

Identifying the correct mechanisms to pass between de Rham cohomology and etale cohomology in a way that is compatible with all of this extra structure is very subtle, and one of the fundamental achievements of Fontaine was really to identify the correct framework in which to phrase the optimal comparison (both in this and also in many related contexts such as crystalline cohomology). (Of course, his work was also instrumental in proving many of these comparison theorems as well.) I think it is fair to say that often the most profound contributions to mathematics come from revealing the underlying structure of what is going on, even if only conjecturally. (To take another random example, take Thurston’s insight into the geometry of 3-manifolds.) Moreover, the reliance of modern arithmetic geometry on these tools can not be overestimated — studying global Galois representations without p-adic Hodge theory would be like studying abelian extensions of $\mathbf{Q}$ without using ramification groups.

Two further points I would be remiss in not mentioning: One sense in which the ring $B_{\mathrm{dR}}$ is “as small as possible” is the amazing conjecture of Fontaine-Mazur which captures which global Galois representations should come from motives. Secondly, Fontaine’s work on all local Galois representations in terms of $(\varphi,\Gamma)$ modules which is crucial even in understanding Motivic Galois representations though p-adic deformations, the fields of norms (with his student Wintenberger, who also sadly died recently), the proof of weak admissibility implies admissibility (with Colmez, another former student, who surprisingly to me only wrote this one joint paper with Fontaine), and the Fargues-Fontaine curve. (I guess this is more than two.)

Probably the first time I talked with Fontaine was at a conference in Brittany (Roscoff) in 2009. That was the first time I ever gave a talk on my work on even Galois representations and the Fontaine-Mazur conjecture, about which Fontaine had some very kind words to say. (One of the most rewarding parts of academia is getting the respect of people you admire.) I never got to know him too well, due (in equal parts) to my ignorance of the French language and p-adic Hodge theory. But he was always a regular presence at conferences at Luminy with his distinct sense of humour. Over a long career, his work continued to be original and deep. He will be greatly missed.

## The classics

I now have the complete collection from of light satirical music of the ’50s and ’60s from the two masters of the form from either side of the pond:

They are both similar and very different at the same time — Lehrer is definitely the more acerbic of the pair, as evidenced by the following pair of quotes concerning satire (themselves satirical):

When Kissinger won the Nobel peace prize, satire died.

The purpose of satire, it has been rightly said, is to strip off the veneer of comforting illusion and cosy half-truth. And our job, as I see it … is to put it back again!”

## Tips for new postdocs

In my role as junior hiring chair, I’ve been thinking a little bit about how a (R1) institution can best serve its postdocs. Many find the transition from graduate student life to being a postdoc somewhat of a rude shock. At the same time as the intellectual support structure of your advisor and fellow graduate students is taken away, while at the same time you have to take on significant teaching responsibilities. Even for those will a fellowship to offset their teaching, it can be a little daunting to figure out exactly how to interact with your new research group.

What should the expectations of a new postdoc be? Many universities assign research mentors to new postdocs, but (in practice) this is essentially meaningless unless it carries with it certain expectations for mentor and mentee to interact. How much of the role should senior faculty help in suggesting problems for postdocs to work on? No doubt the answer to many of these questions is “it depends on the postdoc” but I would love to hear personal stories (positive and negative) about your postdoc experiences, especially as it relates to practical steps that an institution can make to improve the experience.

Feel free to leave your comment anonymously (well, people feel free to do that anyway). I don’t particularly trust my own experience since I feel that I was probably more independent than most as a graduate student, and was fairly happy working alone in my office (not to mention already having a number of collaborations ongoing with Kevin Buzzard and Matthew Emerton). Harvard was a welcoming and friendly place (to me), but my best interactions happened serendipitously more often than not. The initial seeds of my collaboration with Barry started by joining in conversations he was having with Romyar Sharifi and William Stein in front of their offices (all on the 5th floor I believe) discussing (early forms of) Sharifiology in the context of Barry’s paper on the Eisenstein ideal. I had a few lunches with Richard Taylor at the law school (I have a vague memory that I realized this was possible from Toby — could that be right?). Richard is definitely generous with his time, and (in this context) he was ideal for bouncing off ideas. On the other hand, I don’t think Richard’s style in mathematical conversation is to be very speculative; he certainly never suggested any particular problem to me but nor did I ask. My collaboration with Nathan surely started out by virtue of the fact that we would chat socially at tea time.

I can’t quite distill from my own experiences either any recommendations for new postdocs or specific recommendations for institutions (particularly the University of Chicago) to put things in place to improve the lives of postdocs. But perhaps you can help!

Posted in Mathematics | | 4 Comments

## 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.)

## 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!

## 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.

Posted in Uncategorized | | 4 Comments

## And then there were 8:

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

Posted in Mathematics | | 3 Comments