## This year in Jerusalem

I just returned from spending almost a week in Jerusalem (my first ever visit to Israel). The main reason for my visit was to talk with Alex Lubotzky (and Shai Evra) about mathematics, but there was plenty of time for other mathematical activities — I gave a four hour talk on cohomology to computer scientists, chatted with Kazhdan (and also Akshay), caught up with Alex Gamburd, Ehud de Shalit, and went to the presentation of the Ostrowski medal to Akshay Venkatesh (with a virtual laudation by Peter Sarnak).

But this post is less about the mathematics (hopefully more on that when the theorems are proved), but rather my other (mostly culinary) adventures.

My first night out, I was curious how Ethiopian food in Israel would compare to Chicago. However, my taxi driver had other ideas, and instead took me to a Kurdish restaurant (Ima) where I ended up with a pretty nice lamb dish. While walking home from dinner, I stumbled across the Chords Bridge (the “Bridge of Strings”). My particular approach presented me with a visual paradox: the bridge appeared to be straight with a central column with cables to either side. These cables appeared as lines sweeping out a ruled surface. Since the bridge was straight, these two surfaces should essentially have formed one surface, but they appear to meet at right angles at the column, which made no sense. Since my description also probably doesn’t make so much sense, I took a video:

(Admittedly my geometrical intuition is not so great, but I couldn’t work out what was going on until I saw it again from a different angle.)

Tuesday morning was my “time off” as a tourist. I think the old city would have been much better with a local guide, but I mostly just wandered around between ancient sites and an infinite number of tchotchke shops. Next stop was Machane Yehuda market, and lunch at the hippest restaurant in town. The shikshukit was delicious:

Next stop was the fanciest coffee in Jerusalem (not that good)

Akshay and I went back to the market for dinner and had the shamburak at Ishtabach along with some pretty good local beer.

On Wednesday, I was contemplating going straight back to the hotel and going to sleep after an undergraduate lecture by Akshay (full jetlag mode at this point, the talk itself was great). But then I ran into Alex Gamburd, who suggested going out to dinner and said he knew of a place which sold food from “biblical times.” At that point, my spirits were instantly lifted, and there was no choice about what I was going to do. So we jumped into a taxi and off we went to Eucalyptus, to have (amongst other things) chicken stuffed in figs (yes, I thought that was just a poor english translation for figs stuffed in chicken, but no, chicken stuffed in figs). The owner came out to chat with us, and claimed that this dish had won a special prize in Melbourne and had also appeared in Vogue (I couldn’t verify these claims, but they were tasty).

A few more things en passant:

A “reception” at Hebrew University apparently does not include Champagne, much for the worse for anyone who had to suffer though my subsequent basic notions seminar. (Hat tip to Michael Schein for telling me this in advance.)

Here’s Alex Gamburd and Andre Reznikov arguing over a point concerning Stalin:

Near the old city:

The campus appears to be overrun by cats. Well, overrun is an exaggeration, but then saying the campus is “run by cats” conveys a somewhat different image (which may or may not be accurate).

## The paramodular conjecture is false for trivial reasons

(This is part of a series of occasional posts discussing results and observations in my joint paper with Boxer, Gee, and Pilloni mentioned here.)

Brumer and Kramer made a conjecture positing a bijection between isogeny classes of abelian surfaces $A/\mathbf{Q}$ over the rationals of conductor N with $\mathrm{End}_{\mathbf{Q}}(A) = \mathbf{Z}$ and paramodular Siegel newforms of level N with rational eigenvalues (up to scalar) that are not Gritsenko lifts (Gritsenko lifts are those of Saito-Kurokawa type). This conjecture is closely related to more general conjectures of Langlands, Clozel, etc., but its formulation was made more specifically with a view towards computability and falsifiability (particularly in relation to the striking computations of Poor and Yuen).

The recognition that the “optimal level” of the corresponding automorphic forms is paramodular is one that has proved very useful both computationally and theoretically. Moreover, it is almost certain that something very close to this conjecture is true. However, as literally stated, it turns out that the conjecture is false (though easily modifiable). There are a few possible ways in which things could go wrong. The first is that there are a zoo of cuspidal Siegel forms for GSp(4); it so happens that the forms of Yoshida, Soudry, and Howe–Piatetski-Shapiro type never have paramodular eigenforms (as follows from a result of Schmidt), although this depends on the accident that the field $\mathbf{Q}$ has odd degree and no unramified quadratic extensions (and so the conjecture would need to be modified for general totally real fields). Instead, something else goes wrong. The point is to understand the relationship between motives with $\mathbf{Q}$-coefficients and motives with $\overline{\mathbf{Q}}$-coefficients which are invariant under the Galois group (i.e. Brauer obstructions and the motivic Galois group.)

It might be worth recalling the (proven) Shimura-Taniyama conjecture which says there is a bijection between cuspidal eigenforms of weight two with rational eigenvalues and elliptic curves over the rationals. Why might one expect this to be true from general principles? Let us imagine we are in a world in which the Fontaine-Mazur conjecture, the Hodge conjecture, and the standard conjectures are all true. Now start with a modular eigenform with rational coefficients and level $\Gamma_0(N).$ Certainly, one can attach to this a compatible family of Galois representations:

$\displaystyle{\mathcal{R} = \{\rho_p\}, \qquad \rho_p: \mathrm{Gal}(\overline{\mathbf{Q}}/\mathbf{Q}) \rightarrow \mathrm{GL}_2(\overline{\mathbf{Q}}_p).}$

with the property that the characteristic polynomials $P_q(T) = 1 - a_q T + q$ of Frobenius at any prime $q \nmid Np$ have integer coefficients, and the representations are all de Rham with Hodge-Tate weights [0,1]. But what next? Using the available conjectures, one can show that there must exist a corresponding simple abelian variety $E/\mathbf{Q}$ which gives rise to $\mathcal{R}.$ The key to pinning down this abelian variety is to consider its endomorphism algebra over the rationals. Because it is simple, it follows that the endomorphism algebra is a central simple algebra $D/F$ for some number field F. From the fact that the coefficients of the characteristic polynomial are rational, one can then show that the number field F must be the rationals. But the Albert classification puts some strong restrictions on endomorphism rings of abelian varieties, and the conclusion is the following:

Either:

1. $E/\mathbf{Q}$ is an elliptic curve.
2. $E/\mathbf{Q}$ is a fake elliptic curve; that is, an abelian surface with endomorphisms over $\mathbf{Q}$ by a quaternion algebra $D/\mathbf{Q}.$

The point is now that the second case can never arise; the usual argument is to note that there will be an induced action of the quaternion algebra on the homology of the real points of A, which is impossible since the latter space has dimension two. (This is related to the non-existence of a general cohomology theory with rational coefficients.) In particular, we do expect that such modular forms will give elliptic curves, and the converse is also true by standard modularity conjectures (theorems in this case!). A similar argument also works for all totally real fields. On the other hand, this argument does not work over an imaginary quadratic field (more on this later). In the same way, starting with a Siegel modular form with rational eigenvalues whose transfer to GL(4) is cuspidal, one should obtain a compatible family of irreducible 4-dimensional symplectic representations $\mathcal{R}$ with cyclotomic similitude character. And now one deduces (modulo the standard conjectures and Fontaine-Mazur conjecture and the Hodge conjecture) the existence of an abelian variety A such that:

Either:

1. $A/\mathbf{Q}$ is an abelian surface.
2. $A/\mathbf{Q}$ is a fake abelian surface; that is, an abelian fourfold with endomorphisms over $\mathbf{Q}$ by a quaternion algebra $D/\mathbf{Q}.$

There is now no reason to suspect that fake abelian surfaces cannot exist. Taking D to be indefinite, the corresponding Shimura varieties have dimension three, and they have an abundance of points — at least over totally real fields. But it turns out there is a very easy construction: take a fake elliptic curve over an imaginary quadratic field, and then take the restriction of scalars!

You have to be slightly careful here: one natural source of fake elliptic curves comes from the restriction of certain abelian surfaces of GL(2)-type over $\mathbf{Q},$ and one wants to end up with fourfolds which are simple over $\mathbf{Q}.$ Hence one can do the following:

Example: Let $B/\mathbf{Q}$ be an abelian surface of GL(2)-type which acquires quaternion multiplication over an imaginary quadratic field K, but is not potentially CM. For example, the quotient of $J_0(243)$ with coefficient field $\mathbf{Q}(\sqrt{6})$ with $K = \mathbf{Q}(\sqrt{-3}).$ Take the restriction to K, twist by a sufficiently generic quadratic character $\chi,$ and then induce back to $\mathbf{Q}.$ Then the result will be a (provably) modular fake abelian surface whose corresponding Siegel modular form has rational eigenvalues. Hence the paramodular conjecture is false.

Cremona (in his papers) has discussed a related conjectural correspondence between Bianchi modular forms with rational eigenvalues and elliptic curves over K. His original formulation of the conjecture predicted the existence of a corresponding elliptic curve over K, but one also has to allow for fake elliptic curves as well (as I think was pointed out in this context by Gross). The original modification of Cremona’s conjecture was to only include (twists of) base changes of abelian surfaces of GL(2)-type from Q which became fake elliptic curves over K, but there is no reason to suppose that there do not exist fake elliptic curves which are autochthonous to K, that is, do not arise after twist by base change. Indeed, autochthonous fake elliptic curves do exist! We wrote down a family of such surfaces over $\mathbf{Q}(\sqrt{-6}),$ for example. (We hear through Cremona that Ciaran Schembri, a student of Haluk Sengun, has also found such curves.) On the other hand, the examples coming from base change forms from Q have been known in relation to this circle of problems for 30+ years, and already give (by twisting and base change) immediate counter-examples to the paramodular conjecture, thus the title.

It would still be nice to find fake abelian surfaces over $\mathbf{Q}$ (rather than totally real fields) which are geometrically simple. I’m guessing that (for D/Q ramified only at 2 and 3 and a nice choice of auxiliary structure) the corresponding 3-fold may be rational (one could plausibly prove this via an automorphic form computation), although that still leaves issues of fields of rationality versus fields of definition. But let me leave this problem as a challenge for computational number theorists! (The first place to look would be Jacobians of genus four curves [one might be lucky] even though the Torelli map is far from surjective in this case.)

Let me finish with one fake counter example. Take any elliptic curve (say of conductor 11). Let $L/\mathbf{Q}$ be any Galois extension with Galois group $Q,$ the quaternion group of order 8. The group $Q$ has an irreducible representation $V$ of dimension 4 over the rationals, which preserves a lattice $\Lambda.$ If you take

$A = E^4 = E \otimes_{\mathbf{Z}} \Lambda,$

then $A$ is a simple abelian fourfold with an action of an order in $D,$ (now the definite Hamilton quaternions) and so gives rise to compatible families $\mathcal{R}$ of 4-dimensional representations which are self-dual up to twisting by the cyclotomic character. However, the four dimensional representations are only symplectic with respect to a similitude character which is the product of the cyclotomic character and a non-trivial quadratic character of $\mathrm{Gal}(L/\mathbf{Q}),$ and instead they are orthogonal with cyclotomic similitude character. So these do not give rise to counterexamples to the paramodular conjecture. A cursory analysis suggests that the quaternion algebra associated to a fake abelian surface which gives rise to a symplectic $\mathcal{R}$ with cyclotomic similitude character should be indefinite.

## Hiring Season

Lizard 1: Wait, explain again why we bury our young in the sand and thereby place them into mortal peril?

Lizard 2: So they develop character! If it was good enough for me, it’s good enough for them.

(Feel free to choose your own metaphors.)

Posted in Mathematics, Politics, Rant, Travel | | 1 Comment

## Abandonware

For a young mathematician, there is a lot of pressure to publish (or perish). The role of for-profit academic publishing is to publish large amounts of crappy mathematics papers, make a lot of money, but at least in return grant the authors a certain imprimatur, which can then be converted into reputation, and then into job offers, and finally into pure cash, and then coffee, and then back into research. One great advantage of being a tenured full professor (at an institution not run by bean counters) is that I don’t have to play that game, and I can very selective in what papers I choose to submit. In these times — where it is easy to make unpublished work available online, either on the ArXiv, a blog, or a webpage — there is no reason for me to do otherwise. Akshay and I are just putting the finishing touches on our manuscript on the torsion Jacquet–Langlands correspondence (a project begun in 2007!), and approximately 100 pages of the original version has been excised from the manuscript. It’s probably unlikely we will publish the rest, not because we don’t think its interesting, but because it can already be found online. (Although we might collect the remains into a supplemental “apocrypha” to make referencing easier.) Sarnak writes lots of great letters and simply posts them online. I wrote a paper a few years ago called “Semistable modularity lifting over imaginary quadratic fields.” It has (IMHO) a few interesting ideas, including one strategy for overcoming the non-vanishing of cohomology in multiple degrees in an $l_0 = 1$ situation, a way of proving a non-minimal modularity lifting theorem in an (admittedly restricted) $l_0 = 1$ situation without having to use Taylor’s Ihara Avoidance or base change (instead using the congruence subgroup property), and an argument explaining why the existence of Nilpotent ideals in Scholze’s Galois representation is no obstruction to the modularity lifting approach in my paper with David. But while I wrote up a detailed sketch of the argument, gave a seminar about it, and put the preprint on my webpage, I never actually submitted it. One reason was that David and I were (at the time, this was written in 2014-2015 or so) under the cosh by an extremely persnickety referee (to give you some idea, the paper was submitted in 2012 and was only just accepted), and I couldn’t stomach the idea of being raked over the coals a second time merely to include tedious details. (A tiny Bernard Woolley voice at the back of my head is now saying: excuse me minister, you can’t be raked over by a cosh, it doesn’t have any teeth. Well done if you have any idea what I am talking about.) But no matter, the paper is on my webpage where anyone can read it. As it happens, the 10 author paper has certainly made the results of this preprint pretty much entirely redundant, but there are still some ideas which might be useful in the future someday. But I don’t see any purpose whatsoever in subjecting an editor, a reviewer, and (especially) myself the extra work of publishing this paper.

So I am all in favor of avoiding publishing all but a select number of papers if you can help it, and blogging about math instead. So take a spoon, pass around the brandy butter and plum pudding, and, for the rest of this post, let us tuck in to something from the apocrypha.

Galois Extensions Unramified Away From One Place:

I learned about one version of this question in the tea room at Havard from Dick Gross. Namely, does there exist a non-solvable Galois extension K/Q unramified at all primes except p? Modular forms (even just restricting to the two eigenforms of level one and weights 12 and 16) provide a positive answer for p greater than 7. On the other hand, Serre’s conjecture shows that this won’t work for the last three remaining primes. Dick explained a natural approach for the remaining primes, namely to consider instead Hilbert modular forms over a totally real cyclotomic extension ramified at p (once you work out how to actually compute such beasts in practice). And indeed, this idea was successfully used to find such representations by Lassina Dembélé in this paper and also this paper (with Greenberg and Voight). But there is something a little unsatisfactory to me about this, namely, these extensions are all ramified at $p$ and $\infty.$ What if one instead asks Gross’ question for a single place?

Minkowski showed there are no such extensions when $v = \{\infty\},$ but I don’t see any obstruction to there being a positive answer for a finite place. The first obvious remark, however, is that Galois representations coming from Hilbert modular forms are not going to be so useful in this case at least when the residual characteristic is odd, for parity reasons.

On the other hand, conjecturally, the Langlands program still has something to say about this question. One could ask, for example, for the smallest prime p for which there exists a Galois representation:

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

whose image is big (say not only irreducible but also not projectively exceptional) and is unramified at all places away from p including infinity. (This is related to my first ever blog post.) Here is how one might go about finding such a representation, assuming the usual suite of conjectures. First, take an imaginary quadratic field F, and then look to see if there is any extra mod-p cohomology of $\mathrm{GL}_2(\mathcal{O}_F)$ in some automorphic local system which is not coming from any of the “obvious” sources. If you find such a class, you could then try to do the (computationally difficult) job of computing Hecke eigenvalues, or alternatively you could do the same computation for a different such imaginary quadratic field E, and see if you find a weight for which there is an “interesting” class simultaneously for both number fields. If there are no such classes for any of the (finitely many) irreducible local systems modulo p, then there are (conjecturally) no Galois representations of the above form.

There are some heuristics (explained to me by Akshay) which predict that the number of Galois representations of the shape we are looking for (ignoring twists) is of the order of 1/p. On the other hand, no such extensions will exist for very small p by combining an argument of Tate together with the Odlyzko bounds. So the number of primes up to X for which there exist such a representation might be expected to be of the form

$\log \log X - \log \log C$

for some constant C to account for the lack of small primes (which won’t contribute by Tate + Odlyzko GRH discriminant bounds). This is unfortunately a function well-known to be constant, and in this case, with the irritating correction term, it looks pretty much like the zero constant. Even worse, the required computation becomes harder and harder for larger p, since one needs to compute the cohomology in the corresponding local system of weight $(k,k)$ for k up to (roughly) p. Alas, as it turns out, these things are quite slippery:

Lemma: Suppose $\overline{\rho}$ is absolutely irreducible with Serre level 1 and Serre weight k and is even. Assume all conjectures. Then:

1. The prime $p$ is at least 79.
2. The weight $k$ is at least 33.
3. If $\overline{\rho}$ exists with $k \le 53,$ then $p > 1000.$
4. If $\overline{\rho}$ exists with $k = 55,$ then $p > 200,$ or $p =163,$ and $\overline{\rho}$ is the unique representation with projective image $A_4.$

Of course the extension for $p = 163$ (which is well-known) does not have big image in the sense described above.
The most annoying thing about this computation (which is described in the apocrypha) is that it can only be done once! Namely, someone who could actually program might be able to extend the computation to (say) $p \le 200,$ but the number of extensions which one would expect to see is roughly $\log \log 200 - \log \log 79,$ which is smaller than a fifth. So maybe an extension of this kind will never be found! (Apologies for ruining it by not getting it right the first time.)

## The ABC conjecture has (still) not been proved

The ABC conjecture has (still) not been proved.

Five years ago, Cathy O’Neil laid out a perfectly cogent case for why the (at that point recent) claims by Shinichi Mochizuki should not (yet) be regarded as constituting a proof of the ABC conjecture. I have nothing further to add on the sociological aspects of mathematics discussed in that post, but I just wanted to report on how the situation looks to professional number theorists today. The answer? It is a complete disaster.

This post is not about making epistemological claims about the truth or otherwise of Mochizuki’s arguments. To take an extreme example, if Mochizuki had carved his argument on slate in Linear A and then dropped it into the Mariana Trench, then there would be little doubt that asking about the veracity of the argument would be beside the point. The reality, however, is that this description is not so far from the truth.

Each time I hear of an analysis of Mochizuki’s papers by an expert (off the record) the report is disturbingly familiar: vast fields of trivialities followed by an enormous cliff of unjustified conclusions. The defense of Mochizuki usually rests on the following point: The mathematics coming out of the Grothendieck school followed a similar pattern, and that has proved to be a cornerstone of modern mathematics. There is the following anecdote that goes as follows:

The author hears the following two stories: Once Grothendieck said that there were two ways of cracking a nutshell. One way was to crack it in one breath by using a nutcracker. Another way was to soak it in a large amount of water, to soak, to soak, and to soak, then it cracked by itself. Grothendieck’s mathematics is the latter one.

While rhetorically expedient, the comparison between Mochizuki and Grothendieck is a poor one. Yes, the Grothendieck revolution upended mathematics during the 1960’s “from the ground up.” But the ideas coming out of IHES immediately spread around the world, to the seminars of Paris, Princeton, Moscow, Harvard/MIT, Bonn, the Netherlands, etc. Ultimately, the success of the Grothendieck school is not measured in the theorems coming out of IHES in the ’60s but in how the ideas completely changed how everyone in the subject (and surrounding subjects) thought about algebraic geometry.

This is not a complaint about idiosyncrasy or about failing to play by the rules of the “system.” Perelman more directly repudiated the conventions of academia by simply posting his papers to the arXiV and then walking away. (Edit: Perelman did go on an extensive lecture tour and made himself available to other experts, although he never submitted his papers.) But in the end, in mathematics, ideas always win. And people were able to read Perelman’s papers and find that the ideas were all there (and multiple groups of people released complete accounts of all the details which were also published within five years). Usually when there is a breakthrough in mathematics, there is an explosion of new activity when other mathematicians are able to exploit the new ideas to prove new theorems, usually in directions not anticipated by the original discoverer(s). This has manifestly not been the case for ABC, and this fact alone is one of the most compelling reasons why people are suspicious.

The fact that these papers have apparently now been accepted by the Publications of the RIMS (a journal where Mochizuki himself is the managing editor, not necessary itself a red flag but poor optics none the less) really doesn’t change the situation as far as giving anyone a reason to accept the proof. If anything, the value of the referee process is not merely in getting some reasonable confidence in the correctness of a paper (not absolute certainty; errors do occur in published papers, usually of a minor sort that can be either instantly fixed by any knowledgeable reader or sometimes with an erratum, and more rarely requiring a retraction). Namely, just as importantly, it forces the author(s) to bring the clarity of the writing up to a reasonable standard for professionals to read it (so they don’t need to take the same time duration that was required for the referees, amongst other things). This latter aspect has been a complete failure, calling into question both the quality of the referee work that was done and the judgement of the editorial board at PRIMS to permit papers in such an unacceptable and widely recognized state of opaqueness to be published. We do now have the ridiculous situation where ABC is a theorem in Kyoto but a conjecture everywhere else. (edit: a Japanese reader has clarified to me that the newspaper articles do not definitively say that the papers have been accepted, but rather the wording is something along the lines of “it is planned that PRIMS will accept the paper,” whatever that means. This makes no change to the substance of this post, except that, while there is still a chance the papers will not be accepted in their current form, I retract my criticism of the PRIMS editorial board.)

So why has this state persisted so long? I think I can identify three basic reasons. The first is that mathematicians are often very careful (cue the joke about a sheep at least one side of which is black). Mathematicians are very loath to claim that there is a problem with Mochizuki’s argument because they can’t point to any definitive error. So they tend to be very circumspect (reasonably enough) about making any claims to the contrary. We are usually trained as mathematicians to consider an inability to understand an argument as a failure on our part. Second, whenever extraordinary claims are made in mathematics, the initial reaction takes into account the past work of the author. In this case, Shinichi Mochizuki was someone who commanded significant respect and was considered by many who knew him to be very smart. It’s true (as in the recent case of Yitang Zhang) that an unknown person can claim to have proved an important result and be taken seriously, but if a similarly obscure mathematician had released 1000 pages of mathematics written in the style of Mochizuki’s papers, they would have been immediately dismissed. Finally, in contrast to the first two points, there are people willing to come out publicly and proclaim that all is well, and that the doubters just haven’t put in the necessary work to understand the foundations of inter-universal geometry. I’m not interested in speculating about the reasons they might be doing so. But the idea that several hundred hours at least would be required even to scratch the beginnings of the theory is either utter rubbish, or so far beyond the usual experience of how things work that it would be unique not only in mathematics, but in all of science itself.

So where to from here? There are a number of possibilities. One is that someone who examines the papers in depth is able to grasp a key idea, come up with a major simplification, and transform the subject by making it accessible. This was the dream scenario after the release of the paper, but it becomes less and less likely by the day (and year). But it is still possible that this could happen. The flip side of this is that someone could find a serious error, which would also resolve the situation in the opposite way. A third possibility is that we have (roughly) the status quo: no coup de grâce is found to kill off the approach, but at the same time the consensus remains that people can’t understand the key ideas. (I should say that whether the papers are accepted or not in a journal is pretty much irrelevant here; it’s not good enough for people to attest that they have read the argument and it is fine, someone has to be able to explain it.) In this case, the mathematical community moves on and then, whether it be a year, a decade, or a century, when someone ultimately does prove ABC, one can go back and compare to see if (in the end) the ideas were really there after all.

Posted in Mathematics, Politics, Rant | | 33 Comments

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 very successful postdocs (not to mention the occasional visitors) — a new era has already begun, with the hiring of Yifeng Liu and Bao Le Hung.

## Abelian Surfaces are Potentially Modular

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:

$\displaystyle{ \zeta_{X,S}(s) = \prod \frac{1}{1 - N(x)^{-s}}}$

where the product runs over closed points of a smooth integral model. From the Weil conjectures, the function $\zeta_{X,S}(s)$ is absolutely convergent for s with real part at least $1+m/2,$ where $m = \mathrm{dim}(X).$ One has the following well-known conjecture:

Hasse–Weil Conjecture: The function $\zeta_{X,S}(s)$ extends to a meromorphic function on the complex plane. Moreover, there exists a rational number A, a collection of polynomials $P_v(T)$ for v dividing S, and infinite Gamma factors $\Gamma_v(s)$ such that

$\displaystyle{ \xi_{X}(s) = \zeta_{X,S}(s) \cdot A^{s/2} \cdot \prod_{v|\infty} \Gamma_v(s) \cdot \prod_{v|S} \frac{1}{P_v(N(v)^{-s})}}$

satisfies the functional equation $\xi_X(s) = w \cdot \xi_X(m+1-s)$ with $w = \pm 1.$

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 $\zeta_{X,S}(s)$ 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 $\zeta_{X,S}(s)$ is (up to some missing Euler factors at S) the Dedekind zeta function $\zeta_F(s)$ of F, and the conjecture is a theorem of Hecke. If X is a curve of genus zero over F, then $\zeta_{X,S}(s)$ is $\zeta_F(s) \zeta_F(s-1),$ 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 $1 + m/2,$ 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 $\mathbf{Q}.$ 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 $\mathbf{Q}(2^{1/4}).$ 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 $\mathrm{GSp}_4(\mathbf{F}_3)$ 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

$\overline{\rho}: G_{\mathbf{Q}} \rightarrow \mathrm{GSp}_4(\mathbf{F}_3)$

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 $\overline{\rho}.$ This immediately reduces to the question of finding rational points on some twist of the moduli space $\mathcal{A}_2(3).$ 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 $\mathrm{Sp}_4(\mathbf{F}_3),$ and that action is not compatible with the birational map to $\mathbf{P}^3,$ so the resulting twist is not a priori a Severi-Brauer variety. However, something quite pleasant happens — there is a degree six cover

$\mathcal{A}^{-}_2(3) \stackrel{6:1}{\rightarrow} \mathcal{A}_2(3)$

(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 $\mathbf{Q},$ 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 $\mathrm{SL}_2(\mathbf{Z}).$ This gives a direct relationship between modular forms and the coherent cohomology of modular curves; namely, cuspidal modular forms of weight two and level $\Gamma_0(N)$ are exactly holomorphic differentials on the modular curve $X_0(N).$ 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 $H_1(X_0(N),\mathbf{R})$ 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.)

Finally, as promised, here are the videos: