The first rule of number theory swim club is…talk about number theory swim club! Anyone can join, as long as you are a number theorist at the University of Chicago with a current gym membership. The standard meeting time is at the Myers-McLoraine Pool on MWF at 11:30, but we work on an honour system here. The following will be updated on a semi-regular basis. Leave a comment if you want to join in.

• George “the shark” Boxer: 2 3 4 5 6 8 9 11 12 visits.
• Frank “the minnow” Calegari: 3 4 5 6 7 8 9 11 12 14 15 visits.
• Matt “iron man metallic hydrogen man” Emerton: 0 1 2 3 4 5 6 visits.
• Joel “mariachi band” Specter (special guest): 1 visit.

## The True Heirs To Ramanujan

If you are in need of some light relief, you could do worse than peruse the opinions of Doron Zeilberger, who, if viewed strictly through the lens of these ramblings, appears to have a relationship with theoretical mathematics something along the lines of the Unabomber’s relationship with technology. (Let me add that my feelings in real life about Zeilberger is that he has proved some amazing theorems, an opinion which I am absolutely sure is not reciprocated.)

Zeilberger’s opinion 151 is somewhat of a doozy, calling out Ken Ono as a member of the “fancy math gang” who “stole” Ramanujan. First, the idea that one could imagine what Ramanujan thought of the modern theory of mock theta functions (or any other part of mathematics influenced by his legacy) is pure BS. Second, once you create something in mathematics, it transcends its creator; what Ramanujan actually would have thought of his legacy is mostly irrelevant. Maybe Robert Langlands thinks we are all fools for not taking up the double-bitted axe and the cross-cut saw and devoting our lives to the trace formula, and maybe he’s right, but in reality the Langlands program will go in a very different direction to what Langlands anticipated and most likely be better for it. Third, it’s hard to maintain any legitimacy making criticisms of the way modern mathematics is done while insisting that you don’t actually know any “fancy” mathematics, whether that is true or not. But finally, and this is what is most amusing about this opinion, is that I am usually inclined to criticise Ken on precisely the same point, except in the exact opposite direction. That is, my frustration with the theory of mock modular forms is not that it’s too fancy, but rather that it’s nowhere near fancy enough! The subject is crying out for a treatment which incorporates representation theory, where “shadows” are related to the (reducible) principal series which has a discrete series quotient, and where the amazing special structures which have been discovered are given a more algebraic framework rather than the subject as it currently stands: a wild west of crazy q-series, Lerch sums, and indefinite theta series. Whether such an approach would actually be useful is hard to guarantee in advance, but the relevance for the classical theory of modular forms cannot be overestimated.

The process of mathematics is as follows. You discover or observe some phenomenon, and you try to explain it. While trying to explain it, you may come up with a more general theory which explains not only the original example, but also an entire family of examples. And usually at this point, the general theory is more interesting than the original example, because it has more explanatory power to explain why things are true. That doesn’t mean the original example is no longer interesting, but it has to be viewed in the more general context. There is room in mathematics for crazy unique examples that don’t fit a pattern and very general theory. But there are no “heirs” to Ramanujan, because mathematics doesn’t work that way. The fact that Ramanujan’s name will always be linked to Deligne (a mathematician of a quite different sort, to say the least) is testament to that.

Posted in Mathematics, Rant | | 8 Comments

## Public Displays of Mathematics

Let me start by saying that I’m in favor of making the effort to both educate the public about mathematics (as well as science more generally) and to convey to them a sense of the excitement of our discipline. But the science always has to come first, and should never be twisted for the purposes of sensationalism. I understand that I have more antipathy than most towards this sort of thing, but I wanted to discuss a few examples in this post of things which I have seen recently that have particularly annoyed me.

Examples of “math in the news” which didn’t quite live up to the hype have been around for a while. There was the whole E8 thing. I’m not sure how these things start, but it has to involve some unholy trinity of sensationalist journalism, self-promoting universities (or institutes), and complicit [or merely naive] authors. It’s not so easy to untangle the web of blame in any particular situation, but, at the very least, let me recommend to anyone to avoid AIM when it comes to matters of publicity. Not even wrong had an discussion on this particular case, but there’s also an interesting follow up on the E8 article here, where, to give credit where it is due, Oliver Roeder somewhat redeems himself for his previous crimes. (To be clear, the math behind the E8 story — removed from any breathless claims about how it will change the word — is pretty interesting.)

Second, let me also admit that I could be completely wrong about all of this. Obviously I’m not the target audience for popular articles on mathematics, and maybe, even when the truth is stretched beyond recognition or even just a little in that direction, such publicity is good for mathematics. I really don’t know. I acknowledge that I can be out of touch on some issues. I am, after all, someone who gets annoyed by the idea of NPR discussing non-classical music. But (at least in that case) I am sufficiently self-aware to appreciate that my opinion on the matter can safely be disregarded. Maybe that’s true in this case as well. Still, I feel that one can write with excitement about mathematics to a general audience and still be faithful to the mathematics. I grew up with a collection of articles reprinted from Scientific American in the ’60s and ’70s, and they were never afraid to challenge their audience with difficult technical concepts in order to elucidate some often difficult but important idea. One can also find such writings nowadays, although almost always in blog form rather than in print. (I don’t really read popular math blogs that much, but this blog is an example of how one can demonstrate the delights of our subject without being super-technical and yet still being honest about the underlying mathematics.)

I have little faith that university publicists or (even worse) journalists have much interest in being accurate. But I save the most opprobrium for mathematicians who make unreasonable claims about their own work. To make it clear, I was previously a little critical and cheeky about the publicity surrounding the LMFDB here, but that wouldn’t even rise to the level of a small misdemeanor compared to the crimes outlined below. I’m taking here about levels of intellectual dishonesty which would be more appropriate at a business school.

Plimpton 322: I was first alerted to this by a recent post on facebook linking to this article. A few sample sentences are as follows:

“This means it has great relevance for our modern world. Babylonian mathematics may have been out of fashion for more than 3,000 years, but it has possible practical applications in surveying, computer graphics and education. This is a rare example of the ancient world teaching us something new.”

At first, I interpreted this as merely the type of quality rubbish one expects from the Guardian. But then, to my horror, I clicked on the accompanying video to discover that this nonsense was coming directly from the authors (mathematicians, not historians) themselves. Due to ignorance, I will leave aside here the historical claims of the authors (which to be fair are the main point of the article, although they also have been met with skepticism by actual historians) and merely comment on a few of their mathematical claims, specifically, that “Perhaps this different and simpler way of thinking has the potential to unlock improvements in science, engineering, and mathematics education today.” This is so patently absurd that it’s not worth spending much time discussing it, but here goes. There is nothing (and indeed far less) in the “exact” nature of the tablet that isn’t immediately a consequence of the usual rational parametrization of the circle. (Yes, you can use slopes instead of angles!) The “sexagesimal” aspect of the tablet is also a red herring. If you take any base B which is not a prime power, it admits a similar tablet comprised of Pythagorean triples with a side length one and the two other sides given by terminating decimals whose ratios (for large enough triples) are dense in any interval. Here’s a fragment of the corresponding decimal version:

(The column consists of triples $(\delta,a,c)$ with $(a,b,c)$ a Pythagorean triple, b an S-unit with S=10, and $\delta = (c/b)^2$ given as an exact decimal. The triples are ordered by $\delta$ and limited by some height restriction.) Since 60 has more distinct prime factors than 10, the size of the entries in this table is little larger, but that’s about it. Of course, even if you wanted to base a primitive trigonometric system on exact ratios, you would much prefer to use rational points on the unit circle of small height, rather than insisting that the ratios involved had finite expansions, which is very restrictive. (I’m not denying that this may have been convenient historically for numerical computation, I’m only addressing the absurd claim that we have something mathematically to learn from this tablet.) I would stop short of saying that the claims of the authors are fraudulent, but I would go way further than to say they are simply overreaching. Let’s stick with saying that they are vastly overstated purely in order to drum up public interest for their own professional enhancement. And this type of irresponsible behavior leads, inevitably, to this:

OK, enough of that nonsense, let us move on.

Numberphile: OK, perhaps this will be a little bit more controversial. Perhaps the correct thing to bear in mind with this rant is to recall my comment about NPR and classical music above: just because it really annoys me doesn’t mean that I can’t simultaneously accept that it’s probably a good idea for NPR to be somewhat inclusive (I guess). Numberphile is funded by MSRI and that’s probably a good thing, but it still (sometimes) annoys the hell out of me. Should anyone care? I’m not sure. It’s also important to note that there are better and worse numberphile videos — if they restricted themselves to the good ones I would only have very positive things to say. Readers may be aware of the infamous 1+2+3+4+5…=-1/12 video (see here for a takedown). But it gets worse. And not necessary worse in the “this is just rubbish” kind of way, but in the “this gives absolutely the wrong impression about what mathematics is and dresses it up as some ridiculously stupid parlour game instead of something with deep and profound connections” kind of way. There’s a lot of dross to draw from, but here is one typical example:

OK, so what’s the problem? A tiny bit of mathematical knowledge reveals that the (concept of) the Mills’ constant is an interesting observation about what we know concerning (upper bounds for) gaps between primes. But that’s not what one gets out of this video at all. At first, it seems as though there is some mysterious prime generating constant — perhaps you as a youtube viewer can discover a closed form and reveal the mystery of the primes! But this is just rubbish, the actual number (or smallest such number) is of little interest. It’s true that they are slightly more honest towards the end of the video, but the actual mathematics behind this story is always completely obscured. Honestly, if they had just spent a little time (maybe even a minute) saying something at least tangentially related to the real point behind Mills’ constant I would have been much happier. Is enthusiasm better than accuracy? (genuine question). To me, Numberphile can sometimes seem to be the video series that will launch a thousand cranks rather than a thousand mathematicians. It doesn’t help that there’s a bit too much emphasis on recreational mathematics in the worst sense (2 million views between them), which are to real mathematics roughly what eating play-doh is to molecular gastronomy. (Hmmm, maybe a bad analogy, I can totally imagine a play-doh course at Alinea.)

It’s not easy to get it right when it comes to publicizing mathematics (and mathematicians), but it can be done (here, here, and here to name three recent examples). But it helps to start with something serious and try to explain how interesting it is.

## MSRI Now

Continuing on the theme of the last post (Buzzard related viral videos), you can now view Buzzard’s MSRI course (in progress at the time of this post) online here. Having previously excoriated MSRI for restricting how many people can attend such workshops, I must now congratulate them on doing an excellent job in the audio-visual department and making the lectures available to everyone. Students at many levels could learn a lot by watching these and making an honest effort to think about the (implicit) exercises. Even if you know the material, it is still fun to watch; a little like your cool uncle telling you a familiar bedside story but with his own subversive twist. For various psychological reasons, I suspect that those watching the videos now as they come out will have a lower dropout rate than those who watch them later. So go watch them now! (Unless you are a student at Brown, of course.)

Kevin is always refreshingly honest about things he was confused by as a student (or is still confused by now), although sometimes it is reminiscent of Volodya “reminding” almost every speaker at the start of his seminar that his is a beginner and so the speaker will have to go very slowly. Along those lines, here are some (very) tangential remarks on the lectures so far.

When I was a student, I always got very confused when someone talked about the “closure” of the commutator subgroup [G,G]. The basic problem was that I couldn’t conceive of taking the quotient of $\widehat{\mathbf{Z}}$ by 1 and getting anything other than the trivial group. Of course that is what you should get unless you are doing it wrong, because anyone who thinks about profinite groups as abstract groups are probably crazy.

That said, here’s an idle question: is the commutator subgroup [W,W] of the the Weil group of a local field K actually already closed? I believe that the corresponding result for the (local) Galois group G itself is positive (essentially as a consequence of the fact that G is a finitely generated pro-finite group), but W has a distinctly non-compact quotient $\mathbf{Z},$ so I’m not sure. Maybe this is an easy question, I don’t know.

Another random fact: I was a graduate student at Berkeley in 2000 when Richard gave a colloquium on the local Langlands conjectures for GL(n). One aspect of the talk I remember was Richard defining the p-adic numbers, to which Mariusz Wodzicki cried out: “excuse me, this is Berkeley, do you really think you need to define the p-adic numbers?.” At this point, someone else cried out “Yes!” and the talk continued as planned. But the part of this story that is relevant here is that I somehow internalized (either at this talk or before) the fact that, long before Harris-Taylor, the local Langlands conjectures had been proved for GL(n) when p > n (which mirrors the story for n = 2). But I was surpised to find out recently (i.e. this week) that this result was not something from the distant past, but rather was a theorem of 1998 from (friend of the blog) Michael Harris in Inventiones.

Posted in Mathematics | | 1 Comment

## Redraw the Balance

This video would be inspiring even if it didn’t star someone that I know:

I suspect that Tamzin’s other youtube video will probably not end up with 20+ million views, being somewhat more…technical, I guess you could say.

Of course, having just watched this, your task is now to imagine what a new incarnation of Doctor who might look like.

Posted in Politics | | 2 Comments

## New Results in Modularity, Part II

This is part two of series on work in progress with Patrick Allen, Ana Caraiani, Toby Gee, David Helm, Bao Le Hung, James Newton, Peter Scholze, Richard Taylor, and Jack Thorne. Click here for Part I

It has been almost 25 years since Wiles first announced his proof of Taniyama-Shimura, and, truthfully, variations on his method have been pretty much the only game in town since then (this paper included). In all generalizations of this argument, one needs to have some purchase on the integral structure of the automorphic forms involved, which requires that they contribute in some way to the cohomology of an arithmetic manifold (locally symmetric space). This is because it is crucial to be able to exploit the integral structure to study congruences between modular forms. Let’s briefly recall Wiles’ strategy. One starts out with a residual representation

$\overline{\rho}: G_S \rightarrow \mathrm{GL}_2(\mathbf{F}_p)$

which one assumes to be modular, that is, is the mod-p reduction of a representation associated to a modular form which is assumed to have some local properties similar to rho. One then considers a deformation ring R which captures all deformations of the residual representation which “look modular” of the right weight and level (some aspects of Serre’s conjecture due to Ribet are empoyed here, although Skinner-Wiles came up with a base change trick to circumvent some of these difficulties). On the automorphic side, one looks at the cohomology groups M = H^1(X,Z_p)_m of modular curves (X = X_0(N)) localized at a maximal ideal m of the Hecke algebra T associated to rhobar, and proves that there is a surjective map:

$R \rightarrow \mathbf{T}_{\mathfrak{m}}.$

Already many deep theorems have been used to arrive at this point. To begin, one needs Galois representations associated to modular forms, but moreover, one needs to know that these representations satisfy all of the expected local-global compatibilities at the primes in S. In the case of modular forms, all of these facts were basically known before Wiles.

The next step, which lies at the heart of the Taylor-Wiles method, is to introduce certain auxiliary sets Q of carefully chosen primes, and consider the spaces M_Q = H^1(X_1(Q),Z_p)_m which relate to spaces of modular forms of larger level. If T_Q is the associated Hecke algebra, and R_Q is the corresponding deformation ring in which ramification is allowed not only at S but now also at Q, there are compatible maps as follows:

The key point concerning how one chooses the sets Q is to ensure that, even though R_Q may get bigger, its infinitesimal tangent space does not. Hence all the R_Q are quotients of some fixed ring R_oo = Z_p[[X_1,…,X_q]]. (Here q is chosen so that q = |Q|.) In this process, all the rings also have an auxiliary action of a ring S_oo = Z_p[[T_1,…,T_q]] of diamond operators, coming from the Galois group of X_1(Q) over X_0(Q) on the automorphic side, and the inertia groups at Q on the Galois side. The action of S_oo on these modules factors through R_Q by construction, by local global compatibility at primes dividing Q. After throwing away the Galois representations almost entirely (but keeping the diamond operators), one can patch the modules M_Q/p^n for different sets of primes Q, and arrive at a patched module M_oo for R_oo and S_oo such that:

• The module $M_{\infty}$ has positive rank as an $S_{\infty}$ module.
• If $\mathfrak{a}$ is the augmentation ideal of $S_{\infty},$ then $R_{\infty}/\mathfrak{a} = R,$ and $M_{\infty}/\mathfrak{a} = M.$

The first statement may be viewed as saying that there are “lots” of automorphic forms. On the other hand, the fact that R_oo has the same dimension of S_oo says that there are not “too many” Galois representations. Indeed, this friction is enough in this context to prove that M_oo is free over R_oo, and then to deduce the same claim for M over R, from which R = T follows. (Already included here is a innovation due to Diamond where one deduces freeness as a consequence rather than building it in as an assumption.) The argument I have very briefly sketched above is really only a proof of modularity in the minimal case. The general case requires a completely separate argument to bootstrap from minimal to non-minimal level using two further ingredients: Wiles’ numerical criterion, and a lower bound on the congruence ideal necessary to apply the numerical criterion, which ultimately follows from Ihara’s Lemma.

The “first generation” of improvements to Wiles consisted of understanding enough integral p-adic Hodge theory to make the required arugments on the Galois side. Notable papers here include the work of Conrad-Diamond-Taylor and Breuil-Conrad-Diamond-Taylor (but let us also not forget here the contribution of The Hawk). Improvements along these lines continue to today, and are very closely interwined with p-adic Langlands program and work of Breuil, Colmez, Kisin, Emerton, Paškūnas, and many others.

The “second generation” of improvements consisted of relaxing the assumption that R_oo is smooth, by allowing instead R_oo to have multiple components (but still of the same dimension) associated to different components in the local deformation rings at primes in S (at p and away from p). This innovation was due to Kisin, who also introduced the notion of framing to handle this.

The “third generation” of improvements (somewhat orthogonal to the second) cames from replacing 2-dimensional representations with n-dimensional representations, but still under some very restrictive assumptions on the image of rho. One key consequence of these assumptions is that the spaces of modular forms M_Q = H^*(X_1(Q),Z_p)_m all occur inside a single cohomology group, which allows one to control the growth of these spaces when patching. Here one thinks of the work of Clozel-Harris-Taylor. Also pertinent is that the analog of Ihara’s Lemma is open for higher rank groups; Taylor came up with a technique to bypass it when proving modularity lifting theorems now known as “Ihara avoidance.”

(Of course there were many other developments less directly relevant to this post, including but not limited to Skinner-Wiles and Khare-Wintenberger.)

The problem with considering general representations for GL(n) for n > 2, even over Q, is that the automorphic forms are spread over a number of different cohomology groups, in fact in some range [q_0,q_0 + 1, … ,q_0 + l_0] for specific invariants q_0 and l_0.
This manifests itself in two ways:

1. There are not enough automorphic forms; the patched modules M_oo will not be free over S_oo.
2. There are not enough Galois representations: the ring R_oo does not have the same dimension as S_oo but rather dim R_oo = dim S_oo – l_0.

Of course these problems are related! My work with David Geraghty was precisely about showing how to make these problems cancel each other out. The rough idea is as follows. The cohomology groups H^*(X_1(Q),Z_p)_m which contain interesting classes in characteristic zero occur in the range [q_0,…,q_0+l_0]. Suppose one knows this to be true integrally as well, even with coefficients over F_p instead of Z_p. Then instead of patching the cohomology groups M_Q themselves, one instead patches complexes P_Q of length l_0. The result is a complex P_oo of finite free S_oo modules of length l_0, with an action of R_oo on the cohomology of this complex. But the only way the cohomology of this complex can be small enough to admit an action of R_oo is if the complex is a free resolution of the patched module M_oo of cohomology groups in the extreme final degree, and moreover it also implies that M_oo is big enough as in Wiles’ original argument to give an R=T theorem. Note that it is crucial here that one work with the torsion in integral cohomology. It is quite possible that, at all auxiliary levels Q, there are no more automorphic forms at level Q than are were at level 1. (This can only happen for l_0 > 0, and the idea that torsion should be a suitable replacement is the moral of my paper with Barry Mazur.) These argument is also compatible with the improvements to the method including Taylor’s “Ihara Avoidance” argument.

On the other hand, there is a big problem. This argument required many inputs which were completely unknown at the time we worked this out, so our results were very conditional. To be precise, our results were conditional on the following desiderata:

1. The existence of Galois representations on Hecke rings T which acted as endomorphisms of H^*(X,Z/p^nZ) for locally symmetric spaces X associated to GL(n)/F.
2. The stronger claim that the Galois representations constructed in part 1 satisfied the correct “local-global” compatibility statements for all v in S (including v dividing p).
3. The vanishing of the cohomology groups H^i(X,Z/p^nZ)_m outside the range i in [q_0,…,q_0+l_0], for a non-Eisenstein ideal m.

A different approach to some of these questions (which Matt and I discussed, see here) involves first passing to completed cohomology, where one expects (or hopes!) that all the cohomology groups except in degree q_0 should vanish after localization at a non-maximal ideal.

The first big breakthrough was the result of Scholze, who proved part 1 above, at least up to issues concerning a nilpotent ideal (this was discussed previously on this blog). Another innovation appeared in Khare-Thorne, where it was observed that one can sometimes drop the third assumption under the strong condition that there existed global automorphic forms with the exact level structure corresponding to the original representation. (Unfortunately, in the l_0 > 0 setting, there is no way to produce such forms.)

So this is roughly where we stood in 2016. The key new ingredient which led to this project was the new result of Caraiani and Scholze proving vanishing theorems for the cohomology of non-compact Shimura varieties in degrees above the middle dimension (localized at m) under the assumption of certain genericity hypotheses on m. Since the cohomology of the boundary (for suitably chosen Shimura varieties) is precisely related to the cohomology of arithmetic locally symmetric spaces for GL(n) over CM fields, this allowed for the first time a new construction of the Galois representations for GL(n) which directly related them to the Galois representations coming from geometry. (I say “directly related,” but perhaps I mean simply more direct than Peter’s original construction.) In particular, it was clear to Caraiani and Scholze that this result should have implications for the required local-global compatibility result above. Meanwhile, the IAS had just started a new series of workshops on emerging topics. I guess that Richard must have had conversations with Ana about her work with Peter, which led them to choosing this as the theme, namely:

Ana Caraiani and Peter Scholze are hopeful of extending the methods of their joint paper arXiv:1511.02418 to non-compact Shimura varieties. This would give a new way to attack local-global compatibility at p for some of the Galois representations Scholze attached to torsion classes in the cohomology of arithmetic locally symmetric spaces. The aim of this workshop will be to understand how much local-global compatibility can be proved and to explore the consequences of this, particularly for modularity questions.

So now (1) was available, there was an approach to (2), and a technique for avoiding (3). One issue with the Khare-Thorne trick, however, was that it involved localizing at some prime ideal of characteristic zero, and so did not interact so well with Ihara Avoidance, which was crucial for any sort of applicable theorem. Here’s the subtely, which can be described even in the case when l_0 = 0. The usual Ihara avoidance game is to compare deformation rings R and R’ at Steinberg level and ramified principal series level respectively (after making a base change to ensure that the prime v at the relevant prime q satisfies N(v) = 1 mod p). Let M and M’ be the corresponding modules. One has that M/p = M’/p and R/p = R’/p. Suppose, however, that M behaved perfectly as expected, so that M_oo was free (even of rank one say) over S_oo and free over R_oo. What could happen, if one doesn’t have vanishing of cohomology outside a single degree, is that M’_oo/p = M_oo/p is free over S_oo/p, but that M’_oo is the cohomology of a non-trivial complex S_oo —> S_oo given by multiplication by p. So M’_oo is trivial in characteristic zero, even though M’_oo/p = M_oo/p. So this is a problem. But it is exactly a problem which was resolved during the workshop. The point, very loosely speaking, is that even though the complexes “S_oo” and “S_oo –>[p]—> S_oo” have the same H^0 after reducing modulo p and taking cohomology, their intersection with S_oo/p are quite different on the derived level, so if one can formulate a version of derived Ihara avoidance, then one is in good shape.

So what remained? First, there were a number of technical issues, some of which could be dealt with individually, and one had to make sure that all the fixes were compatible. For example, it is straightforward to modify the original strategy in my paper with David to handle the issue of only having Galois representations up to nilpotence ideals of fixed nilpotence, but one had to make sure this would not interfere with the more subtle derived Ihara avoidance type arguments. Relevant here was the work of Newton and Thorne which placed some of the arguments with complexes more naturally in the derived category. Second, there was the issue of really proving local-global compatibility from the new results of Caraiani-Scholze. A particularly interesting case here was the ordinary case. The rough problem one has to deal with here is deducing that rho is ordinary from knowing that $\rho \oplus \rho^{\vee}$ is ordinary. But be careful — the latter representation is reducible and so really a pseudo-representation — so it’s not even clear what ordinary this means (though see work of Wake and Wang Erickson, as well as of my student Joel Specter). It turns out that some interesting and subtle things turn up in this case which were found by the “team” of people who wrote up this section. (Although we acheived quite a lot in a week, there were obviously a list of details to be worked out, and we divided ourselves up into certain groups to work on each part of the paper.) But I think we were fairly confident at this point that everything would work out. What was my role in the writing up process you ask? I was selected as the ENFORCER, who goes around harassing everybody else to work and write up their sections of the paper while sipping on Champagne. Presumably I was less selected for my organizational skills and more for my ablity to tell Richard Taylor what to do.

So there we have it! It was clear even during the workshop that some improvements to our arguments were possible, but since the paper is already going to be quite long, we did not try to be completely comprehensive. I expect a number of improvements will follow shortly. I would not be surprised to see in a few years a modularity result for regular weight compatible systems over CM fields which are as complete as the ones (say) in BLGGT.

Finally, I should mention that while the paper is almost completely written, the usual caveats apply about work in progress which has not been completely written up (although we are almost done…)

## New Results In Modularity, Part I

I usually refrain from talking directly about my papers, and this reticence stems from wishing to avoid any appearance of tooting my own horn. On the other hand, nobody else seems to be talking about them either. Moreover, I have been involved recently in a number of collaborations with multiple authors, thus sufficiently diluting my own contribution enough to the point where I am now happy to talk about them.

The first such paper I want to discuss has 9(!) co-authors, namely Patrick Allen, Ana Caraiani, Toby Gee, David Helm, Bao Le Hung, James Newton, Peter Scholze, Richard Taylor, and Jack Thorne. The reason for such a large collaboration is a story of itself which I will explain at the end of the second post. But for now, you can think of it as a polymath project, except done in a style more suited to algebraic number theorists (by invitation only).

In this first post, I will start by giving a brief introduction to the problem. Then I will state one of the main theorems and give some (I hope) interesting consequences. In the next post, I will be a little bit more precise about the details, and explain more precisely what the new ingredients are.

Like all talks in the arithmetic of the Langlands program, we start with:

The Triangle:

Let F be a number field, let p be a prime, and let S be a finite set of places containing all the infinite places and all the primes above p. Let G_S denote the absolute Galois group of the maximal extension of F unramified outside S. In many talks in the Langlands program, one encounters the triangle, which is a conjectural correspondence between the following three objects:

• A: Irreducible pure motives M/F (with coefficients) of dimension n.
• B: Continuous irreducible n-dimensional p-adic representations of G_S (for some S) which are de Rham at the places above p.
• C: Cuspidal algebraic automorphic representations $\pi$ of $\mathrm{GL}(n)/F.$

In general, one would like to construct a map between any two of these objects, leading to six possible (conjectural) maps, which we can describe as follows:

• A->B: This is really the only map we understand, namely, etale cohomology. (I’m being deliberately vague here about what a motive actually is, but whatever.)
• B->A: This is the Fontaine-Mazur conjecture, and maybe some parts of the standard conjectures as well, depending on exactly what a motive is.
• B->C: This is “modularity.”
• C->B: This is the existence of Galois representations associated to automorphic forms.
• A->C: We really think of this as A->B->C and also call this modularity.
• C->A: Again, this is a souped up version of C->B. But note, we still don’t understand how to do this even in cases where C->B is very well understood. For example, suppose that $\pi$ comes from a Hilbert modular form with integer coefficients of trivial level over a totally real field F of even degree. We certainly have an associated compatible family of Galois representations, and we even know that its symmetric square is geometric. But it should come from an elliptic curve, and we don’t know how to prove this. The general problem is still completely open (think Maass forms). On the other hand, often by looking in the cohomology of Shimura varieties, one proves C->A and uses this to deduce that C->B.

This triangle is also sometimes known as “reciprocity.” The other central tenet of the Langlands program, namely functoriality, also has implications for this diagram. Namely, there are natural operations which one can easily do in case B which should then have analogs in C which are very mysterious.

Weight Zero: For all future discussions, I want to specialize to the case of “weight zero.” On the motivic/Galois side, this corresponds to asking that the representations are regular and which Hodge-Tate weights which are distinct and consecutive, namely, [0,1,2,…,n-1]. The hypotheses that the weights are distinct is a restrictive but crucial one — already the case when F = Q and the Hodge-Tate weights are [0,0] is still very much open (specifically, the case of even icosahedral representations). On the automorphic side, the weight zero assumption corresponds to demanding that the $\pi$ in question contribute to the cohomology of the associated locally symmetric space with constant coefficients.

For example, if n=2, then we are precisely looking at abelian varieties of GL(2) type over F (e.g. elliptic curves). This is an interesting case! We know they are modular if F is Q, or even a quadratic extension of Q. More generally, we know that if F is totally real, then such representations are at least potentially modular, that is, their restriction to some finite extension $F'/F$ is modular. This is often good enough for many purposes. For example, it is enough to prove many cases of (some version of) B->A. In this case, we have quite complete results, although still short of the optimal conjectures, especially in the case when the residual representation is reducible.

There are many other modularity lifting results generalizing those for n=2, but they really involve Galois representations whose images have extra symmetry properties. In particular, they are either restricted to representations which preserve (up to scalar) some orthogonal or symplectic form, or they remain unchanged if one conjugates the representation by an outer automorphism of G_F (for example when $F/F^+$ is CM and one conjugates by complex conjugation). There were basically no unconditional results which applied either in the situation that n > 2 or that F was not completely real, and the representation did not otherwise have some restrictive condition on the global image. Our first main theorem is to prove such an unconditional result. Here is such a theorem (specialized to weight zero):

Theorem [ACCGHLNSTT]: Let F be either a CM or totally real number field, and p a prime which is unramified in F. Let

$\rho: G_S \rightarrow \mathrm{GL}_n(\overline{\mathbf{Q}_p})$

be a continuous irreducible representation which is crystalline at v|p with Hodge-Tate weights [0,1,..,n-1]. Suppose that

1. The residual representation $\overline{\rho}$ has suitably big image.
2. The residual representation is “modular” in the sense that there exists an automorphic form $\pi_0$ for $\mathrm{GL}(n)/F$ of weight zero and level prime to p such that $\overline{r}(\pi_0) = \overline{\rho}.$

Then $\rho$ is modular, that is, there exists an automorphic representation $\pi$ of weight zero for $\mathrm{GL}(n)/F$ which is associated to $\rho.$

One could be more precise about what it means to have big image. In fact, I can do this by saying that it has enormous image after restriction to the composite of the Galois closure of F with the pth roots of unity. Here enormous is a technical term, of course. There is also a version of this theorem with an ordinary (rather than Fontaine-Laffaille) hypothesis (more on this next time).

Let me now give a few nice theorems which can be deduced from the theorem above:

Theorem [ACCGHLNSTT]: Let E be an elliptic curve over a CM field F. Then E is potentially modular.

When I had a job interview at MIT in 2006, I was asked by Michael Sipser, the chair at the time, to come up with a theorem which (in a best case scenario) I would hope to prove in 10 years. I said that I wanted to prove that elliptic curves over imaginary quadratic fields were modular. (Reader, I got the job … then went to Northwestern.) It is very gratifying indeed that, roughly 10 years later, this result has actually been proved and that I have made some contribution towards its eventual resolution. (OK, we have potential modularity rather than modularity, but that is splitting hairs…). It is also amusing to note that a number of co-authors were still in high school at this time! (Fact Check: OK, just one…)

In fact, one can improve on the theorem above:

Theorem [ACCGHLNSTT]: Let E be an elliptic curve over a CM field F. Then Sym^n(E) is potentially modular for every n. In particular, the Sato-Tate conjecture holds for E.

Finally, for an application of a different type, suppose that $\pi$ is a weight zero cuspidal algebraic automorphic representation for $\mathrm{GL}(2)/F.$ For each prime v of good reduction, one can associate to $\pi_v$ a pair of Satake parameters $\{\alpha_v,\beta_v\}$ satisfying $|\alpha_v \beta_v| = N(v).$ The Ramanujan conjecture says that one has

$|\alpha_v| = |\beta_v| = N(v)^{1/2}.$

An equivalent formulation is that the sum $a_v$ of these two eigenvalues satisfies $|a_v| \le 2 N(v)^{1/2}.$ We prove the following:

Theorem [ACCGHLNSTT]: Let F be a CM field, and let $\pi$ be a weight zero cuspidal algebraic automorphic representation for $\mathrm{GL}(2)/F.$ Then the Ramanujan conjecture holds for $\pi.$

If F is totally real, then the Ramanujan conjecture follows from Deligne’s theorem. One can associate to $\pi$ a motive, whose Galois representation is either $\rho = \rho(\pi)$ or $\rho^{\otimes 2}.$ Then, by applying purity to these geometric representations, one deduces the result. (Of course, this was famously proved by Deligne himself in the case when F = Q. The case of a totally real field, especially in cases where one has to go via a motive assoicated to $\rho^{\otimes 2},$ is due (I think) to Blasius.) This is decidedly not the way we prove this theorem. In fact, we do not know how to prove the Fontaine-Mazur conjecture for the representation $\rho$ associated to $\pi,$ even in the weak sense of showing that $\rho$ or even $\rho^{\otimes 2}$ appears inside the cohomology of some projective variety. Instead, we prove that $\mathrm{Sym}^n \rho$ is potentially modular, then use the weaker convexity bound to prove the inequality:

$|\alpha_v|^n \le N(v)^{n/2 + 1/2}.$

Taking n sufficiently large, we deduce that $|\alpha_v| \le N(v)^{1/2},$ which (by symmetry) proves the result. Experts will recognize this as precisely Langlands’ original strategy for proving Ramanujan using functoriality! In a certain sense, this is the first time that Ramanujan has been proved without a direct recourse to purity. I say “in some sense”, because there is also the ambiguous case of weight one modular forms. Here the Ramanujan conjecture (which is $|a_p| \le 2$ in this case) was deduced by Deligne and Serre as a consequence of showing that $\rho$ has finite image so that alpha_v and beta_v are roots of unity. On the other hand, that argument does also simultaneously imply that the representations are motivic. So our theorem produces, I believe, the only cuspidal automorphic representations for $\mathrm{GL}(n)/F$ for which we know to be tempered everywhere and yet for which we do not know are directly associated in any way to geometry.

Question: Suppose I’m sitting in my club, and Tim Gowers asks me to say what is really new about this paper. What should I say?

Answer: The distinction (say) between elliptic curves over imaginary quadratic fields and real quadratic fields, while vast, is quite subtle to explain to someone who hasn’t thought about these questions. You could explain it, but the club is hardly a place to do so. Instead, go with this narrative: We generalize Wiles’ modularity results for 2-dimensional representations of Q to n-dimensional representations of Q. If you are pressed on previous generalizations, (especially those due to Clozel-Harris-Taylor), say that Wiles is the case GL(2), Clozel-Harris-Taylor is the case GSp(2n), and our result is the case GL(n).

If you had slightly more time, and the port has not yet arrived, you might also try to explain how the underlying geometric objects involved for GSp(2n) are all algebraic varieties (Shimura varieties), but for GL(n) they involve Riemannian manifolds which have no direct connection to algebraic geometry. Here is a good opportunity to name drop Peter Scholze, and explain how this is the first time that the methods of modularity have been combined with the new world of perfectoid spaces.