This last summer, I undertook my last official activity as a faculty member at Northwestern University, namely, graduation day! (I had a 0% courtesy appointment for two years until my last Northwestern students graduated.)

Here I am with four of my six former students. (Richard and Vlad actually graduated in 2016, but were hooded together with Joel in 2017.)

From left-to-right: Richard Moy is a postdoc at Wilamette College in Portland (for previous blog posts on Richard’s work, see Hilbert Modular Forms Part II and Part III), Zili Huang (Thurston and Random Polynomials) has a real job at a consulting firm in Chicago but swung by to say hello on graduation day, Vlad Serban (The Thick Diagonal) has as postdoctoral position in Vienna, and Joel Specter (Hilbert Modular Forms Part II and … hmmm, I guess I didn’t blog about any of his other papers) has just started a postdoc position at Johns Hopkins. Missing are Zoey Guo (Abelian Spiders), now at the Institute of Solid Mechanics at Tsinghua University in Beijing , and my first student Maria Stadnik (who just moved to Florida Atlantic University, and whose thesis predates this blog).

It’s easy to get the sense as a student that math departments are fairly static (which is mostly true over the 4 years or so it takes to do a PhD), but as time goes on, people end up moving around much more than you expect, and the characters of various departments change quite a bit. A sign of good hiring is that your faculty leave because they have been recruited elsewhere! And even though my departure two years ago brought one era of number theory at Northwestern to an end — starting with Matt, then me, two one-year cameo appearances by Toby, and a string of verysuccessfulpostdocs (not to mention the occasionalvisitors) — a new era has already begun, with the hiring of Yifeng Liu and Bao Le Hung.

Today I wanted (in the spirit of this post) to report on some new work in progress with George Boxer, Toby Gee, and Vincent Pilloni.

Recal that, for a smooth projective variety X over a number field F unramified outside a finite set of primes S, one may write down a global Hasse-Weil zeta function:

where the product runs over closed points of a smooth integral model. From the Weil conjectures, the function is absolutely convergent for s with real part at least where One has the following well-known conjecture:

Hasse–Weil Conjecture: The function extends to a meromorphic function on the complex plane. Moreover, there exists a rational number A, a collection of polynomials for v dividing S, and infinite Gamma factors such that

satisfies the functional equation with

Naturally, one can be more precise about the conductor and the factors at the bad primes. In the special case when F = Q and X is a point, then is essentially the Riemann zeta function, and the conjecture follows from Riemann’s proof of the functional equation. If F is a general number field but X is still a point, then is (up to some missing Euler factors at S) the Dedekind zeta function of F, and the conjecture is a theorem of Hecke. If X is a curve of genus zero over F, then is and one can reduce to the previous case. More generally, by combining Hecke’s results with an argument of Artin and Brauer about writing a representation as a virtual sum of induced characters from solvable (Brauer elementary) subgroups, one can prove the result for any X for which the l-adic cohomology groups are potentially abelian. This class of varieties includes those for which all the cohomology of X is generated by algebraic cycles.

For a long time, not much was known beyond these special cases. But that is not to say there was not a lot of progress, particularly in the conjectural understanding of what this conjecture really was about. The first huge step was the discovery and formulation of the Taniyama-Shimura conjecture, and the related converse theorems of Weil. The second was the fundamental work of Langlands which cast the entire problem in the (correct) setting of automorphic forms. In this context, the Hasse-Weil zeta functions of modular curves were directly lined to the L-functions of classical weight 2 modular curves. More generally, the Hasse-Weil zeta functions of all Shimura varieties (such as Picard modular surfaces) should be linked (via the trace formula and conjectures of Langlands and Kottwitz) to the L-functions of automorphic representations. On the other hand, these examples are directly linked to the theory of automorphic forms, so the fact that their Hasse-Weil zeta functions are automorphic, while still very important, is not necessarily evidence for the general case. In particular, there was no real strategy for taking a variety that occurred “in nature” and saying anything non-trivial about the Hasse-Weil zeta function beyond the fact it converged for real part greater than which itself requires the full strength of the Weil conjectures.

The first genuinely new example arrived in the work of Wiles (extended by others, including Breuil-Conrad-Diamond-Taylor), who proved that elliptic curves E/Q were modular. An immediate consequence of this theorem is that Hasse-Weil conjecture holds for elliptic curves over Q. Taylor’s subsequent work on potentially modularity, while not enough to prove modularity of all elliptic curves over all totally real fields, was still strong enough to allow him to deduce the Hasse-Weil conjecture for any elliptic curve over a totally real field. You might ask what have been the developments since these results. After all, the methods of modularity have been a very intense subject of study over the past 25 years. One problem is that these methods have been extremely reliant on a regularity assumption on the corresponding motives. One nice example of a regular motive is the symmetric power of any elliptic curve. On the other hand, if one takes a curve X over a number field, then h^{1,0} = h^{0,1} = g, and the corresponding motive is regular only for g = 0 or 1. The biggest progress in automorphy of non-regular motives has actually come in the form of new cases of the Artin conjecture — first by Buzzard-Taylor and Buzzard, then in the proof of Serre’s conjecture by Khare-Wintenberger over Q, and more recently in subsequent results by a number of people (Kassaei, Sasaki, Pilloni, Stroh, Tian) over totally real fields. But these results provide no new cases of the Hasse-Weil conjecture, since the Artin cases were already known in this setting by Brauer. (It should be said, however, that the generalized modularity conjecture is now considered more fundamental than the Hasse-Weil conjecture.) There are a few other examples of Hasse-Weil one can prove by using various forms of functoriality to get non-regular motives from regular ones, for example, by using the Arthur-Clozel theory of base change, or by Rankin-Selberg. We succeed, however, in establishing the conjecture for a class of motives which is non-regular in an essential way. The first corollary of our main result is as follows:

Theorem [Boxer,C,Gee,Pilloni] Let X be a genus two curve over a totally real field. The the Hasse-Weil conjecture holds for X.

It will be no surprise to the experts that we deduce the theorem above from the following:

Theorem [BCGP] Let A be an abelian surface over a totally real field F. Then A is potentially modular.

In the case when A has trivial endomorphisms (the most interesting case), this theorem was only known for a finite number of examples over In each of those cases, the stronger statement that A is modular was proved by first explicitly computing the corresponding low weight Siegel modular form. For example, the team of Brumer-Pacetti-Tornaría-Poor-Voight-Yuen prove that the abelian surfaces of conductors 277, 353, and 587 are all modular, using (on the Galois side) the Faltings-Serre method, and (on the automorphic side) some really quite subtle computational methods developed by Poor and Yuen. A paper of Berger-Klosin handles a case of conductor 731 by a related method that replaces the Falting-Serre argument by an analysis of certain reducible deformation rings.

The arguments of our paper are a little difficult to summarize for the non-expert. But George Boxer did a very nice job presenting an overview of the main ideas, and you can watch his lecture online (posted below, together with Vincent’s lecture on higher Hida theory). The three sentence version of our approach is as follows. There was a program initiated by Tilouine to generalize the Buzzard-Taylor method to GSp(4), which ran into technical problems related to the fact that Siegel modular forms are not directly reconstructible from their Hecke eigenvalues. There was a second approach coming from my work with David Geraghty, which used instead a variation of the Taylor-Wiles method; this ran into technical problems related to the difficulty of studying torsion in the higher coherent cohomology of Shimura varieties. Our method is a synthesis of these two approaches using Higher Hida theory as recently developed by Pilloni. Let me instead address one or two questions here that GB did not get around to in his talk:

What is the overlap of this result with [ACCGHLNSTT]? Perhaps surprisingly, not so much. For example, our results are independent of the arguments of Scholze (and now Caraiani-Scholze) on constructing Galois representations to torsion classes in Betti cohomology. We do give a new proof of the result that elliptic curves over CM fields are potentially modular, but that is the maximal point of intersection. In contrast, we don’t prove that higher symmetric powers of elliptic curves are modular. We do, however, prove potentially modularity of all elliptic curves over all quadratic extensions of totally real fields with mixed signature, like The common theme is (not surprisingly) the Taylor-Wiles method (modified using the ideas in my paper with David Geraghty).

What’s new in this paper which allows you to make progress on this problem? George explains this well in his lecture. But let me at least stress this point: Vincent Pilloni’s recent work on higher Hida Theory was absolutely crucial. Boxer, Gee, and I were working on questions related to modularity in the symplectic case, but when Pilloni’s paper first came out, we immediately dropped what we were doing and started working (very soon with Pilloni) on this problem. If you have read the Calegari-Geraghty paper on GSp(4) and are not an author of the current paper (hi David!), and you look through our manuscript (currently a little over 200 pages and [optimistically?!] ready by the end of the year), then you also recognize other key technical points, including a more philosophically satisfactory doubling argument and Ihara avoidance in the symplectic case, amongst other things.

So what about modularity? Of course, we deduce our potential modularity result from a modularity lifting theorem. The reason we cannot deduce that Abelian surfaces are all modular, even assuming for example that they are ordinary at 3 with big residual image, is that Serre’s conjecture is not so easy. Not only is not a solvable group, but — and this is more problematic — Artin representations do not contribute to the coherent cohomology of Shimura varieties in any setting other than holomorphic modular forms of weight one. Still, there are some sources of residually modular representations, including the representations induced from totally real quadratic extensions (for small primes, at least). We do, however, prove the following (which GB forgot to mention in his talk, so I bring up here):

Proposition [BCGP]: There exist infinitely modular abelian surfaces (up to twist) over Q with End_C(A) = Z.

This is proved in an amusing way. It suffices to show that, given a residual representation

with cyclotomic similitude character (or rather inverse cyclotomic character with our cohomological normalizations) which has big enough image and is modular (plus some other technical conditions, including ordinary and p-distinguished) that it comes from infinitely many abelian surfaces over Q, and then to prove the modularity of those surfaces using the residual modularity of This immediately reduces to the question of finding rational points on some twist of the moduli space And this space is rational! Moreover, it turns out to be a very famous hypersurface much studied in the literature — it is the Burkhardt Quartic. Now unfortunately — unlike for curves — it’s not so obvious to determine whether a twist of a higher dimensional rational variety is rational or not. The problem is that the twisting is coming from an action by and that action is not compatible with the birational map to so the resulting twist is not a priori a Severi-Brauer variety. However, something quite pleasant happens — there is a degree six cover

(coming from a choice of odd theta characteristic) which is not only still rational, but now rational in an equivariant way. So now one can proceed following the argument of Shepherd-Barron and Taylor in their earlier paper on mod-2 and mod-5 Galois representations.

What about curves of genus g > 2?: Over there is a tetrachotomy corresponding to the cases g = 0, g = 1, g = 2, and g > 2. The g = 0 case goes back to the work of Riemann. The key point in the g = 1 case (where the relevant objects are modular forms of weight two) is that there are two very natural ways to study these objects. The first (and more classical) way to think about a modular form is as a holomorphic function on the upper half plane which satisfies specific transformation properties under the action of a finite index subgroup of This gives a direct relationship between modular forms and the coherent cohomology of modular curves; namely, cuspidal modular forms of weight two and level are exactly holomorphic differentials on the modular curve On the other hand, there is a second interpretation of modular forms of weight two in terms of the Betti (or etale or de Rham) cohomology of the modular curve. A direct way to see this is that holomorphic differentials can be thought of as smooth differentials, and these satisfy a duality with the homology group by integrating a differential along a loop. And it is the second description (in terms of etale cohomology) which is vital for studying the arithmetic of modular forms. When g = 2, there is still a description of the relevant forms in terms of coherent cohomology of Shimura varieties (now Siegel 3-folds), but there is no longer any direct link between these coherent cohomology groups and etale cohomology. Finally, when g > 2, even the relationship with coherent cohomology disappears — the relevant automorphic objects have some description in terms of differential equations on locally symmetric spaces, but there is no longer any way to get a handle on these spaces. For those that know about Maass forms, the situation for g > 2 is at least as hard (probably much harder) than the notorious open problem of constructing Galois representations associated to Maass forms of eigenvalue 1/4. In other words, it’s probably very hard! (Of course, there are special cases in higher genus when the Jacobian of the curve admits extra endomorphisms which can be handled by current methods.)

Job season is upon us. Now is probably a good time to give applicants (and letter writers!) a few pointers. Of course, there are many other sources of advice on this topic, so let me try to narrow the focus on suggestions that you might not find elsewhere.

But first, I am contractually obligated (and also happy) to remind you all to make sure all your best graduate students (in all fields) apply for a Dickson Instructorship at Chicago. Occasionally people get the impression that our deadline is November 1st. In fact, that is merely the date after which we are allowed to start reading recommendations. In reality, committee members will most likely start reading the files over Thanksgiving break, so definitely try to have all your materials (and letters of recommendation) submitted by then. In contrast, some of the public schools (including the UC system, correct me if I’m wrong) have hard application deadlines. In those cases, it is vital that you submit your application before the deadline (it doesn’t need to be complete, just submitted).

I’m applying (or writing a letter) for the second year in a row. Any tips? A number of people apply when they have an extra year remaining in their current position to a limited number of schools. I don’t know enough game theory to evaluate this strategy, but the scales are definitely tipped in favor of doing this when two body problems are involved. But be warned! There is a technical issue on mathjobs which arises which you almost definitely will not be able to anticipate as an applicant. It is the following. When a letter writer submits a letter of recommendation to mathjobs, there is a default setting on how long that letter can be viewed. And for some ridiculous reason, that time period is something like 18 months. A letter writer can, and I do, change the default period to any date one wants (I usually make the letter expire sometime during the following summer). But not all of your letter writers seem to realize this! That means that when you go to apply the following year, your mathjobs listing will have your letters from the current year AND your letters from the previous year, unless your letter writer actively makes the effort to delete the old letter. The first thing this signals to those reading your letters is that you applied the previous year. This on its own is not so bad. However, it is very often the case that the letter in year N+1 is pretty much identical to the letter in year N. And that does give the impression that the applicant hasn’t really done anything in the previous twelve months. The worst aspect of this problem is that there is not really any way for the candidate prevent it, beyond warning their letter writers about the problem. So this is mostly a reminder for letter writers who are writing for the second time in two years: make sure you delete/replace your letters from the previous year! (Or do make sure your secretaries do this on your behalf, if that’s how you roll.)

Should I write to people at universities letting them know about my application? This is generally considered a worthwhile thing to do, because, even in cases in which you are not offered the job, it does give a way of letting people know about your research. In the other direction, a suitably customized and genuine email can let the relevant people know that you might accept a position if you are offered one. A few caveats, however. I appreciate letters which let me know about an application but don’t require a response. Secondly, there should be some synergy between your own research and the person you are writing to, otherwise it looks a little like you are just spamming everyone. Finally, there should be something at least slightly realistic about your application, especially for more senior positions. (But slightly is good enough.)

How many letters do I really need? Let’s specialize now to the case of postdoc applications, although some of this also applies to tenure track letters. This definitely a case where “more” is usually not “better.” Counting the teaching letter separately, a first approximation would be as follows:

Four shalt thou not count, neither count thou two, excepting that thou then proceed to three. Five is right out.

Here’s the problem with having (say) six letters. Most of the time, as a graduate student, there are not going to be six people who know your thesis work really well. Maybe you feel your application looks a little fancier because Professor Fancy McBoatface agreed to write for you, even though you just had that one conversation at a conference. But then the first letter people will click on will be from Professor McBoatface, which will say something like “I chatted with X at a conference once, it seems like they are doing something interesting, although I don’t know the work very well.” Basically, too many letters will dilute the message. Of course, it does look good if you can get a strong letter from a well known expert who is not at your university, but that is much more likely to happen if you have had some genuine sustained mathematical interaction with that person, rather than some fleeting interaction. (I had letters out of graduate school from Kevin Buzzard, with whom I was writing a paper, and René Schoof, who visited Berkeley for a semester and with whom my interaction was directly related to part of my thesis.) There are circumstances in which there is someone (say your advisor) who has to write for you, but for some reason you suspect that their letter may not be as strong as you would like; that’s one justifiable reason to hedge with an extra letter. But in the end, the people who are going to write the strongest letters for you are probably going to be the people who know your work the best.

I talked previously about work of Wake and Wang-Erickson on deformations of Eisenstein residual representations. In that post, I also mentioned a paper of Emmanuel Lecouturier who has also proved some very interesting theorems. Today, I wanted to talk about some complementary results by my student Eric Stubley in collaboration with Karl Schaefer (a student of Matthew Emerton). To duplicate slightly from that previous post, recall that Matt and I proved the following:

Theorem Let p > 3 be prime, and let N = 1 mod p be prime. If the rank of the cuspidal Hecke algebra of level localized at the Eisenstein prime is greater than one, then

has non-cyclic p-class group. Using work of Merel, one can dispense with the discussion of Hecke algebras and instead give an equivalent reformulation of the first condition, namely, if and only if is a p-th power, where

We followed up this result with the comment:

We expect (based on the numerical evidence) that the condition that the class group of K has p-rank [at least] two is equivalent to the existence of an appropriate group scheme, and thus to [the rank being greater than one].

As noted previously, there are counter-examples, already for p = 7 and N = 337. However, there was still clearly some relationship between these quantities beyond the one-way implication above. In particular, the numerical evidence still stubbornly supported the hope that the converse may indeed be true for p = 5. This is the first theorem that Schaefer and Stubley prove. More precisely, they completely determine the rank of the class group of for primes N which are 1 mod 5.

Theorem [Schaefer, Stubley]: Let be prime. Then the 5-rank r_K of the class group of is either 1, 2, or 3. Moreover:

if and only if the Merel invariant is not a perfect 5th power.

if and only if is a perfect 5th power, and is not a perfect 5th power modulo N.

if and only if and are both 5th powers modulo N.

This also answers a conjecture of Lecouturier. Their argument greatly clarified (to me) the exact relationship between the class group of K and a number of other related quantities in this picture. To recall, a third reformulation of whether the Hecke algebra has non-trivial deformations can be given (as in Wake–Wang-Erickson) by whether a certain pairing between specific classes and in and vanish or not. The point is that the vanishing of a cup product ensures the existence of an extension

and one can show (after some massaging) that c_0 gives rise to something in the p-class group of K. Conversely, if one starts with a class in the p-class group of K, and then takes the Galois closure over Q, then (sometimes) one arrives with a Galois extension M/Q with a Galois representation to GL(3) of the above form. The problem is, in other circumstances, one arrives at a representation which has a much larger Galois group and a map to the Borel subgroup in higher dimension, which looks something like this:

Suppose one now tries to construct a representation of this form in order to find a non-trivial class in the p-class group of K. First, one can start by finding a suitable class which cups trivially with The vanishing of a generalized Merel invariant (under a regularity hypothesis) is exactly what guarantees the existence of such a suitable class at least when m is odd. However, one is then faced with an increasing sequence of obstruction problems in order to climb the ladder and get all the way to the full representation of the form above. Here one has to deal with not only cup products, but also (implicitly) higher Massey products. Ultimately, the relation between the quantity and the deformation rings of Hecke algebras is most precise only when . It turns out that there is still something one can say for however. Consider the higher Merel invariant

for odd values of n. Suppose that p is a regular prime. One can show that if , then at least one of these quantities M_n is a perfect pth power for an odd When p = 5, this is a weaker version of the theorem above. So an optimistic variation on the conjecture above is that if and only if is a perfect pth power of for at least one odd The description of the relationship between these classes (which also come up in Lecouturier, they arise via an explicit analysis of Gauss sums and Stickelberger’s theorem) suggests that this conjecture is too optimistic in general, and indeed there are counter-examples for p = 11. But, Schaefer and Stubley do prove the following:

Theorem [Schaefer, Stubley]: Let p = 7, and let N = 1 mod p be prime. Then the 7-class group of has rank if and only if either M_1 or M_3 is a perfect 7th power modulo N.

For example, consider the previous “counter-example” for N = 337 and p = 7. Here the non-trivial class group is explained by the fact that M_3 is a perfect 7th power modulo N.

One thing I especially like about this result is that there are three groups of people (Wake–Wang-Erickson, Lecouturier, and Schaefer–Stubley) are all working around a similar problem, but their results are complementary to each other. I believe that all five people will be at the upcoming IAS workshop, so I hope to hear more about this then.

I found the following documentary remarkable and quite interesting. Without offering here any opinion on its merits, I certainly give it credit for taking an unpopular position and sticking with it. This blog is no stranger to challenging perceived wisdom, although I usually aim to be slightly more subtle (some may argue I do not always succeed). Here is an excerpt from the opening:

The fishing village of Aldeburgh, home and inspiration to Benjamin Britten, England’s finest 20th century composer, or so it’s widely claimed. In fact, much of what he wrote in the sycophantic, closed world of Aldeburgh was anaemic, and loveless; spiritually dead long before he was buried here in 1976.

I’m not entirely sure what the academic consensus about Britten is nowadays (if any exists). I do appreciate some of his smaller scale choral works. I wouldn’t say that Britten’s work is played excessively in relation to its merit in the US, but possibly things are different in London.

I previously mentioned that I once made (in a footnote) the false claim that for a 11-dimensional representation V of the Mathieu group M_12, the 120 dimensional representation Ad^0(V) was irreducible. I had wanted to write down representations W of large dimension n such that Ad^0(W) of dimension n^2 – 1 was irreducible. In the comments, Emmanuel Kowalski pointed to a paper of Katz where he discusses actual examples (including the 1333 dimensional representation of the Janko group J_4). On the other hand, I recently learned from Liubomir Chirac’s thesis:

that it’s an open problem to determine whether there exists such a representation for all n (although he does write down infinitely many examples in prime power dimension). Chirac’s thesis also lead me to the paper of Magaard, Malle, and Tiep, who do classify all such examples for (central extensions of) simple groups. Turns out that I could have used M_12 after all, or rather the 10-dimensional representation of the double cover 2.M_12, which does have the required property (the 99-dimensional representation factors through M_12, naturally).

One reason (amongst many) that (either of the) 11-dimensional representations V of M_12 do not have Ad^0(V) irreducible is that they are self-dual (oops). On the other hand, if you eyeball the character table, you will find that there is an irreducible representation W of dimension 120. Moreover, let me write down the characters of [V \otimes V^*] – [1] and [W]:

These seem surprisingly close to me! So now the question is, as one ranges over (some class perhaps all) finite groups G, what is the minimum number of conjugacy classes for which

\chi = [V \otimes V^*] – [1] – [W]

can be non-zero for irreducible V and W, assuming that it is non-zero? Since V is irreducible, by Schur’s Lemma, this virtual representation is orthogonal to [1] (unless [W] = [1] which would be silly). So which certainly implies that there must be at least two non-zero entries of opposite signs. I don’t see any immediate soft argument which pushes that bound to 3. I admit, this is a slightly silly question. But still, a beer to anyone who proves the example above is either optimal or comes up with an example with only two non-zero terms. (To avoid silliness, say that the dimension of V has to be at least 5.) The characters above look strikingly similar to me, and it does make we wonder if there is any reason for why they are so close. Perhaps if I knew more about groups, I could feel more confident in just chalking up the resemblance above to a law of small numbers.

Probably a more sensible question is to ask for how small the number of non-zero entries of of [V]-[W] can be for two distinct irreducibles. That question has surely been studied!

I found out a good way to describe how long my commute is: about three minutes more than the length of the second movement of Beethoven’s 9th (the greatest movement!)

On the other hand, that measure proved inaccurate the very next day, when I also found out the answer to “is the drawbridge on Lake Shore Drive ever used?”

(channelling my inner Stanley Kubrick with a little well-timed help from 98.7WFMT). The whole opening/closing of the bridge did cause quite some delay, but the process did, in in the end, finish.