Hilbert Modular Forms of Partial Weight One, Part III

My student Richard Moy is graduating!

Richard’s work has already appeared on this blog before, where we discussed his joint work with Joel Specter showing that there existed non-CM Hilbert modular forms of partial weight one. Today I want to discuss a sequel of sorts to that paper, which also forms part of Richard’s thesis (I should note that he already has five publications and will have 7 or 8 papers by the time he graduates.) The starting observation is as follows. Fix a real quadratic field F. From the perspective of Galois representations, the Hilbert modular forms of partial weight one fall under the case \ell_0 = 1 in the notation of my paper with David Geraghty (this is in the context of coherent cohomology). To orient the reader, let us discuss three classes of such forms:

  1. Hilbert modular forms of weight [2k+1,1] for a real quadratic field F.
  2. Regular algebraic cuspidal automorphic forms for \mathrm{GL}(3)/\mathbf{Q}.
  3. Regular algebraic cuspidal automorphic forms for \mathrm{GL}(2)/F for an imaginary quadratic field F.

Suppose one fixes a tame level N and then looks at the space of such forms as the weights vary. In both of the latter cases, the problem has been raised (or even conjectured, for N = 1 and \mathrm{GL}(3) by Ash and Pollack here), of whether all but finitely many such forms arise via functoriality from a smaller group. More explicitly, one can ask whether:

  1. If G = \mathrm{GL}(2)/F, then all but finitely many cuspidal regular algebraic forms of conductor N either arise (up to twist) via base change from \mathrm{GL}(2)/\mathbf{Q}, or are induced from a quadratic CM extension E/F.
  2. If G = \mathrm{GL}(3)/\mathbf{Q}, then all but finitely many cuspidal regular algebraic forms of conductor N arise up to twist as the symmetric square of a form from \mathrm{GL}(2)/\mathbf{Q}.

Naturally enough, one can make the same conjecture whenever \ell_0 > 0, appropriately formulated. There does not seem to be any case of this conjecture which is known, although there are analogous results (where one fixes the weight and varies the level) in both weight one (where it is almost trivial) and for imaginary quadratic fields (in the work of Calegari-Dunfield and Boston-Ellenberg). Still, the conjectures in varying weight seem pretty hard even for N = 1. In that context, Richard proves the following nice complementary pair of theorems below. Let F = \mathbf{Q}(\sqrt{7}). The field F has narrow class number 2 and there is a unique odd everywhere unramified quadratic character \chi of G_F with fixed field E = F(\sqrt{-1}).

Theorem I (Moy) Let F and \chi be as above. Every Hilbert modular form over F of weight [2k+1,1] and level N = 1 is CM, and in particular is induced from E.

Theorem II (Moy) Let F and \chi be as above. Let M be a strongly compatible family of two dimensional Galois representations of F with determinant \chi, level N = 1, and Hodge–Tate weights [0,0] and [k,-k]. Then M is induced from E.

Theorem I is almost an immediate consequence of Theorem II, with the caveat that one doesn’t quite have complete local-global compatibility for partial weight one modular forms (though results and methods of Luu, Jorza, and Newton get close). Theorem II on the other hand is a consequence of the following:

Theorem III (Moy) Let F and \chi be as above. Let

\rho: G_F \rightarrow \mathrm{GL}_2(\overline{\mathbf{Q}}_3)

be a continuous irreducible representation with determinant \chi that is unramified at all finite places except for one prime v|3. Then \rho is induced from a character of G_E.

The argument in this case is (roughly) the following. Using a Tate-style argument (with discriminant bounds), one proves that the residual representation \overline{\rho} must have semi-simplification \chi \oplus 1. The restriction of \rho to G_E then has the property that its image is pro-3 and unramified outside the fixed prime v|3. Yet one shows by a class field theory computation that the largest abelian 3-extension unramified outside v|3 is cyclic, which (by consideration of the Frattini quotient) immediately implies that the image of \rho restricted to G_E factors through a cyclic quotient as well, and one is done.

Note that to deduce Theorem I, one first has to prove (using a congruence argument) that at the other prime w|3, either:

  1. The representation \rho is unramified at w,
  2. The representation \rho restricted to D_w has unramified semi-simplification. In particular, the generalized eigenvalues of \mathrm{Frob}_w for \overline{\rho} are both the same.

To finish, one rules out the second possibility by computing all the modular residual representations explicitly by doing computations in low weight (this can ultimately be reduced to a computation on the definite quaternion side, although Richard had to write his own programs to do this since the current magma implementation required trivial character for non-parallel weight.)

It is true that these arguments will not suffice for the more general conjecture, but then, I haven’t seen a viable strategy to prove those conjectures either!

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Ventotene, part I

I recently returned from Ventotene, an Island some 40 miles west of Naples, where I attended a very pleasant conference on Manifolds and Groups. I have several mathematical thoughts on the conference, but for today I will content myself to a brief description of some of the extra-curricular activities related to spending a week on an Italian Island. The participants of the conference were scattered over 5 of the Island’s numerous hotels. My particular hotel, the Hotel Borgo Cacciatori, had the feature that it was roughly a mile walk from either downtown or the conference center (the Island itself is not two miles in length). This was good for two reasons. First, it provided the more sedentary amongst us to get some exercise, secondly, being near the highest point of the island, it afforded a nice view of the ocean:


Finally, on the two evenings with no cloud cover, the walk back to the hotel provided a spectacular view of the Milky Way.

The food options included all manner of interesting melanzane based dishes, some good antipasti:

and, of course, some very tasty pork with crackling:


The end of the week saw the start of the feast of Santa Candida, the Island’s patron saint, which included the ceremonial release of a balloon (of which I have a wonderful video which can not be embedded in this post). Some number-theoretic Proustian character was also notable…

Kowalki's Blog

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The distance of the moon

Having just re-read the delightful Italo Calvino story of this name (which you can read here), I was also entranced by the following animation:

Also: my take on the pronunciation of the name of the narrator: the first f is silent, so it comes out at something like “quwolfque.”

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Music in London

London is a wonderful city, an I can certainly imagine moving there if I suddenly happen to become a Russian oligarch. One thing that is absolutely fantastic is the classical music scene, which is surely the best in the world. On my recent visit, I happened to be visiting the Courtauld gallery when I saw a flier indicating a concert in the museum at 4:30. The attendant didn’t appear to know anything about it, despite the fact that the sign was less than three feet from his desk. (During a brief conversation, I casually suggested the performance might consist of music inspired by the work of Cezanne; I later heard him definitively telling other patrons the same story.)

Now a “free classical music concert” with no other details could encompass a wide range of possibilities. Original compositions, perhaps? First year students playing Vivaldi in order to get in some performance practice? A rumour that the music involved a guitar and a flute was not inspiring. But inspiring it turned out to be. The flautist turned out to be Adam Walker, who, amongst other achievements, has been (since 2009) a principal flute of the LSO. Walker gave a wonderful rendition of Debussy’s “Syrinx,” which for your benefit I link to here:

As if that music was too mainstream for a random Sunday afternoon collection of patrons at an art gallery, the next performer dialed it up a notch with a performance of Henri Dutilleux’s “trois strophs sur le nom de Sacher.” This piece comes from a collection of 12 Cello works commissioned by Rostopovich for the 70th birthday of Paul Sacher. This performance (irritatingly, I can no longer recall the name of the performer) was absolutely riveting, although I have to confess this is music outside my usual comfort zone (to quote from this essay: “The only piece in this set to make use of a scordatura, a deliberate mis-tuning of strings, is Henri Dutilleux’s TROIS STROPHES SURLE NOM DE SACHER. In this piece the ‘cello’s G string is tuned down a halfstep to F sharp, and the C string is tuned down a whole step to B flat.”)
Still, this was exactly live music at it’s best — a chance to discover new music.

It turned out that the actual concerts that I attended the rest of the week (Mendelssohn’s Octet on one occasion and some Bach Violin concertos on the other) didn’t quite live up to the standard of this unexpected but wonderful recital.

Other random (non-math) observations from London: the only drinkable coffee in London is served by Australians working in Australian cafes; Soho turns out to be slightly tricky to navigate if there happens to be 50,000 people there for a pride rally; if you’re going to ride a rental bike from Soho to Kensington, there are possibly better routes than Oxford street; One day I need to schedule my trips to London to coincide with a test match at Lord’s; the HOTTEST JULY DAY IN LONDON EVER was not actually that hot.

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Course Announcement

Although this may be of limited interest, I wanted to announce a topics class that I intend to give this winter. The title of the course is, simply, “thesis problems.” The structure of the course is to devote each week to a different problem which I think would make a good thesis, or, at least, a good “first project.” The topics are sufficiently diverse that each week will be independent of the next. The problems range in difficulty, but they have several features in common, including the fact that I have ideas on how to start all of them, and I have complete solutions to none of them. I have no shortage of possible problems, but I will try my best to select those with the following properties:

  1. The background material you will be required to learn in order to attempt any of these problems will be useful even if you do not ultimately end up solving the problem.
  2. For the harder problems, there will be some low hanging fruit if you are not able to solve the problem.
  3. The problems will tie into the research of other faculty members in the department, so you will have at least two people (and frequently many more) to speak with. Pro-tip for graduate students: talk to faculty other than your advisor! For example, at least one (and possibly two) of the problems would be suitable for students of Benson, and obviously many will be suitable for students of Matt as well.

Part of the reason I am bringing this up now is to ask: has anyone taught a class like this before? Are there any pitfalls (obvious or otherwise) that I should be aware of? This course is a spiritual cousin of the numerous problem sessions which are held at conferences. However, I think that these sessions usually have limited success, in part due to the fact that the problems that come up (while usually interesting) are just too hard. I will be especially conscious therefore to choose problems for which I think real progress can be made, and for which there is somewhere to start. You can think of this class a little like what might happen if you came to my office and asked what it might be like to be my student, except spread over three hours a week for nine weeks or so.

You might ask if I’m worried that hungry postdocs will hoover up the problems like Lunt Hall cockroaches eating crumbs after a wine and cheese. The answer is: not really! I think this class would be a success if I end up with (say) two or three students working on problems related to what I talk about. This will leave many other problems which I would still love people to work on. (Though, of course, if you are a postdoc who does want to think about one of these problems, you should probably tell me.)

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A Coq and Bull Story

Author: Michael Harris.
Title: Mathematics Without Apologies.
Source: I eventually gave up waiting for a complimentary signed copy to be sent to me in the mail, and so borrowed a copy from the Northwestern University library. I’ve read up to the beginning of chapter 5 so far.

Thoughts before reading the book: Rational thought and continental philosophy have always been non-overlapping magisteria in my mind. Who better, then, to bridge the gap than Michael Harris, the left-bank mathematician who had returned stateside (well, Manhattan-side at least).

In Mathematics Without Apologies, Michael Harris addresses what it is to be a mathematician. There are several questions raised in this book which I hope to consider on this blog. However, before we begin any analysis of the actual content, there is an issue that first has to be discussed. In a book so thick with opinions, it is quite extraordinary that the author believes that he can not only play the simultaneous roles of case study and provocateur, but also, at the same time, present himself as completely impartial observer, one who is merely “offering up” a goulash of narratives and cultural insights for our consideration. Michael Harris is not the naive innocent that he pretends to be; to me, the author is about as impartial as Socrates is when he asks Thrasymachus what makes a “just man.” On the other hand, how can one win an argument with the author on how strongly he holds certain opinions? Rather than take a detour to discuss the relationship between author and text, and argue about the extent to which an author can, by writing supplementary blog posts or otherwise, clarify the intended implications (or lack of implications) of his own words, let me offer the following solution: from this point on, we shall denote the author of “Mathematics Without Apologies” by michael harris (lower case). To be clear, michael harris is a chimera born out of Michael Harris’ words and my interpretation of them. Michael Harris is allowed to disagree with my interpretation of what michael harris says, but he is not the ultimate arbiter of deciding what michael harris thinks; that is up to the reader(s). The only further words I will say on this topic are as follows: for a book claiming to offer no apologies, there are an inordinate number of strongly worded disclaimers, for example: “[one] … should not mistake this book for a work of scholarship.” [OK, duly noted. Of course, these are the words of Michael Harris, not michael harris, the savage cultural critic.] Voloch complains here of his frustration of the author’s apparent inability to commit to any position, to which Michael Harris is sympathetic, but I suggest that both of them read closer between the lines to see that michael harris does indeed hold some strong opinions. With that out of the way, let us begin considering actual issues raised by the text. For now, let me restrict myself to explaining and largely agreeing with a single opinion of michael harris. We begin by recalling a distinction made in the book between the philosophy of Mathematics and the philosophy of mathematics. The philosophy of Mathematics is concerned purely with epistemological questions, and should be thought of as the subject whose intellectual lineage goes back to the crisis of foundations, and which tries to explain the meaning of mathematical truth and its relationship to knowledge. The philosophy of mathematics, on the other hand, is concerned with how actual mathematicians think (about mathematics), and is a topic of primary interest to the author. What becomes clear when reading the book is that michael harris believes that the philosophy of Mathematics — and indeed the subject of foundations more generally — has nothing at all useful to say or contribute to the professional lives of mathematicians, by whom I mean people like myself, or people like Michael Harris. For example, harris quotes Jeremy Gray (from Gray’s account of the Foundations Crisis) as follows:

The logicist enterprise, even if it had succeeded, would only have been an account of part of mathematics — its deductive skeleton, one might say…. mathematics, as it is actually done, would remain to be discussed.

Or, to quote michael harris directly (p.67), “Capital-F Foundations may be needed to protect mathematics from the abyss of structureless reasoning, but they are not the source of mathematical legitimacy.” Nothing controversial so far, I think. But harris goes further. In a book thick with quotations, the key to any reading is to identify the villain. And the villain in chapter 3 is definitely Voevodsky, quoted here as follows:

If one really thinks deeply about … [the possibility that the foundations of mathematics are inconsistent] … this is extremely unsettling for any rational mind.

Certainly, by this measure, I am either not in a possession of a rational mind, or not a deep thinker about these questions. If someone told me today that Voevodsky had discovered an inconsistency in ZFC, I would care slightly less than if someone told me the Collatz problem had been solved, and care much less than if someone (trustworthy) told me that a serious error had been found in the proof of cyclic base change. In the first case, I would presume that what ever fix (large or small) to foundations needed to be made, it wouldn’t make any difference to the mathematics that I think about. Whereas in the third case, it would make quite a lot of difference. Underlying all of this, of course, is some assumption (by me) that there is some formulation of mathematics which is consistent. There seems to be some evidence for this, including a several thousand year history of mathematics which has required barely any modification at all in light of whatever logical issues arose in the 19th and 20th centuries. This is not to say that michael harris completely dismisses any efforts to study foundations. However, to his mind, and to mine, the ultimate judgment about such an enterprise should be on its effect on mathematics qua mathematics (“If and when univalent foundations is adopted as a replacement for today’s … foundations, it will probably be … triggered by a demonstration of the new method’s superiority in addressing old problems”, p.65). This ties together with the attitude that both michael harris and I share towards computer assisted proofs, which is somewhere between “who cares” and a general skepticism that it will have any relevance to mathematics as we practice it (caveat: I believe michael harris discusses these issues in later chapters which I have not yet read). Again, to understand what michael harris thinks, it suffices to judiciously select those whom he quotes. Here is another quote by harris on p.66:

When [Benecerraf] limits the articulation of mathematical truth to logic and then complains that the ability of mathematicians to refer has been lost, it is no wonder; it is also no wonder that number theorists and geometers have not borrowed the language of logic to do their work.

(Confusingly, I can’t quite work out where this quote is from: the footnote refers to a 2009 reprint of writings of Herman Weyl (who died in 1955), but the quote above implicitly (in the context of the book) seems to refer to an article of Benecerraf from 1973.) Here michael harris is drawing the following distinction. The view, coming from the philosophy of Mathematics, is that mathematicians are trying to understand some truth, and then to try to decipher what form of truth mathematics actually consists of; whether it be a chain of modus ponens all the way down, or what have you. After all, if the goal of mathematicians is to seek the “truth” (whatever that is), then surely it follows that it is important to put the notion of truth on some firm philosophical footing. Yet the philosophy of mathematics view is really quite different. If you asked me why I do mathematics, I wouldn’t, if I was honest, say that I was seeking “the truth.” My attraction to mathematics (and even analytic philosophy) is that I find it a lot of fun, and moreover completely addictive. Analytic philosophy (of the flavour I enjoy) is fun, because it is a (logical or linguistic) puzzle. For example, the observation:

“Hesperus is Phosphorus” is not a tautology

can be used to negate certain claims about the meaning of proper names and the relationship of the necessary to the a priori. (Or even better, to go Quine rather than Frege, “9 = the number of planets” is not a tautology, not least of which because it is currently false.) But do these exercises actually tell us anything about epistemology, or are they really just some form of enjoyable intellectual exercise? Those questions are beyond my pay grade. But honestly, they don’t even interest me that much. I simply enjoy those puzzles for what they are, rather than lay any claim that they are revealing deep truths about human thought. (To be honest, I think if I really felt that I wanted to understand what words mean, I would become a computational linguist, not an analytic philosopher.) In mathematics, too, I enjoy playing with objects (Galois representations, automorphic forms) that are my stock in trade; the fun and beauty aspects are more compelling than the “search for absolute truth.” No doubt michael harris will have more to say on these topics in later chapters. These issues also remind of a conflict between mathematicians and historians of Mathematics. A colleague once reported to me a conversation (or argument) with a historian of Mathematics who claimed that Galois did not know that A_5 was simple. On the other hand, Serre claims here that Galois not only knew that A_n for n \ge 5 was simple, but also that L_2(p):=\mathrm{PSL}_2(\mathbf{F}_p) was simple (for p \ge 5), and that (moreover) Galois really understood these groups. For example, Serre claims that Galois knew that L_2(p) for p > 3 has a transitive action on p points only for p = 5,7,11. (Exercise for the reader!) Of course, it’s a bit of a running joke in mathematics that everything is really due to Gauss, but I side with Serre here (a bold stance, I know), and moreover think that one of the issues is (or could be) the tendency for a historian of Mathematics to view mathematics through the perspective of a Philosopher of Mathematics rather than of mathematics; to fail to distinguish between a formal proof and a genuine understanding.

Other remarks on the first few chapters: Harris quotes Neil Chriss (p.72 “who chose to forgo a promising future in the Langlands program to work for a … hedge fund”) as saying that “The Glass Bead Game is a favorite novel among my mathematician friends.” (the implication being here that Mathematicians would love to have no responsibility to the outside world). Really? Either those mathematicians completely misunderstood the point of that novel, or I did. It seemed to me that one of the key themes of Hesse’s book was that a complete disconnect between intellectual pursuits and the ultimate responsibilities of humanity towards society was a bad thing; possibly not a surprising message for a book which was published in 1943. On the other hand, the implication of the quote above is that Mathematicians dream of Castalian paradise where they can pursue mathematics unencumbered by the realities of society. Curiously enough, Both Michael Harris and michael harris are entities to whom the theme of Hesse’s book (in my reading) seems to hold some appeal. One certainly gets the sense even from the first few chapters that michael harris is deeply concerned with the interaction between intellectuals and broader society. (I am with harris completely when it comes to his opinions on foundations; I expect to differ on matters related to our role in society.)

(p.123) I think that Tom Stoppard did a better job of demonstrating that the symmetry group of Rosencrantz and Guildenstern is \mathbf{Z}/2 \mathbf{Z}.

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Harassed by Springer

Those of you who have ever submitted a paper to any mathematical journal may have noticed that it’s not a particularly speedy process. Nowadays, even a one year turnaround is nothing out of the ordinary. Thus, I always find it slightly amusing (once the paper is accepted) to receive breathlessly worded emails from the publisher demanding that you “review the proofs within 48 hours” with the (implied) risk that acceptance of your paper might be at risk if you don’t rush to meet their deadline.

What happens if you ignore these emails? Well, it turns out you get follow-up emails: “the message below was sent to you several days ago but we have not yet received your corrections. Please return your proof as soon as possible so as not to delay the publication of your article.” Of course, these emails are subtitled First Reminder, so it’s probably safe to ignore those as well. At this point, I’m kind of curious as to how long this process continues. Maybe they will write a short abstract for me and then publish the paper anyway. Presumably my co-author doesn’t mind (though I don’t want to get my editor in trouble).

This all leads me to the following suggestion for disrupting journal publications: submit your paper to journals, but then when they are accepted, put them on your website with an annotation along the lines of accepted by Journal X (you can even include the original acceptance email from the editor for authenticity purposes). All the imprimatur of the journal system, none of the cost!

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