The spoils of Rio

Congratulations to Akshay and Peter — don’t spend all those loonies at once! (Actually, I’m not sure that’s a grammatically correct usage of the word loonies unless they pay the winners in the form of 15000 coins; possibly the Hawk can correct me.)

One could (perhaps) equally offer commiserations — at this point, such a prize has little to no effect on your professional advancement, but may well seriously increase low-level harassment both from the press and from certain types of graduate student who will now follow you around in a ring at conferences at a respectful distance of two to six feet. (Perhaps they already do.)

Of course, echoing Emmanuel’s sentiments, in addition to not forgetting the collaborators of this year’s Fields medalists, let us especially not forget those people (currently estimated as a set of size one cough cough) who have collaborated with at least two of this year’s winners — you have clearly inspired [edit: adeptly ridden the coattails of] the next generation of mathematicians.

Posted in Mathematics, Politics, Waffle | Tagged , , , , , , | 6 Comments

Update on Sato-Tate for abelian surfaces

Various people have asked me for an update on the status of the Sato-Tate conjecture for abelian surfaces in light of recent advances in modularity lifting theorems. My student Noah Taylor has exactly been undertaking this task, and this post is a summary of his results.

First, let me recall the previous status of this conjecture. An explicit form of this conjecture (detailing all the 52 possible different Sato-Tate groups which could occur for abelian surfaces over number fields — 34 of which occur over Q) was given in a paper of Fité, Kedlaya, Rotger, and Sutherland (I recommend either reading these slides or especially watching this video for the background and some fun animations). Christian Johansson gave proofs of this conjecture over totally real fields in many of the possible cases in which the abelian surface had various specific types of extra endomorphisms over the complex numbers by exploiting modularity results that had been used in the proof of the Sato-Tate conjecture for elliptic curves. Over totally real fields, this left essentially four remaining cases:

  1. The case when the Galois representations associated to A decomposes over a quadratic extension L/F into two representations which are Galois twists of each other, and L/F is not totally real.
  2. The case when the Galois representations associated to A decomposes over a quadratic
    extension L/F into two representations which are not Galois twists of each other, and L/F is CM.
  3. The case when the Galois representations associated to A decomposes over a quadratic
    extension L/F into two representations which are not Galois twists of each other, and L/F is neither totally real nor CM.
  4. The case when the geometric endomorphism ring of A is \mathbf{Z}.

Noah has something to say about each of these cases.

Case 1: Noah completed the proof of Sato-Tate in this case using only the methods from BLGGT, by exploiting the fact that the corresponding two-dimensional representations — while possibly only defined over a field L which need not be totally real or CM — in fact give rise to projective representations which extend to F. By a theorem of Tate, each of these representations can be extended to F after twisting by a character, and so the original 4-dimensional representation looks like the tensor product of a 2-dimensional representation over F (which is potentially modular) and an Artin representation. At this point one is in good shape.

Case 2: The Sato-Tate conjecture is proved in this case. This case required the least amount work, because it is pretty much an immediate consequence of the modularity results proved in the 10-author paper.

If the totally real field is Q, this implies the Sato-Tate conjecture for all abelian surfaces except those of type (4).

Cases 3 & 4: In these cases, one can apply the potentially modularity results proved in my (very close to being finished) paper with Boxer, Gee, and Pilloni. It is too much to expect a full proof of Sato-Tate at this point. However, knowing potential modularity allows one to obtain partial results, similar to those of Serre and Kim-Shahidi for the case of elliptic curves (after Wiles but before Clozel-Harris-Taylor). Here is a sample result:

Theorem (Noah Taylor). Let C be a genus two curve over a totally real field F. Then, for any \epsilon > 0, there exists a positive density of primes \mathfrak{p} (with N(\mathfrak{p}) = p), one has

\displaystyle{\# C(\mathcal{O}/\mathfrak{p}) - p  - 1 >  \left(\frac{2}{3} - \epsilon \right) \sqrt{p}}.

Compare this to the Hasse bounds, which imply that the quantity on the LHS has absolute value at most 4 \sqrt{p}.

Of course this theorem is much weaker than the Sato-Tate conjecture. But even the weaker version of this theorem which says that \#C(\mathbf{F}_p) > p + 1 for infinitely many primes was completely open before such curves were known to be potentially modular. Similarly, I don’t think one can prove the corresponding result for elliptic curves without either using something very close to modularity (in the non-CM case) or the equidistribution theorems of Hecke in the CM case. I think the following example is instructive: take the elliptic curve y^2 = x^3 - x which admits CM by the Gaussian integers. One has a formula for the difference a_p= 1+p-\#E(\mathbf{F}_p) as follows: for a prime which is 1 mod 4, one may write p = a^2 + b^2 uniquely in integers by imposing the additional congruence

(a + b i) \equiv 1 \mod (1 + i)^3.

Then one has the formula a_p = 2a.

The problem then becomes: do there exist infinitely many primes p = 1 mod 4 such that one has a > 0? This seems suspiciously like something that can be proven using Cebotarev, but it is not. The problem is that the infinite places of F = \mathbf{Q}(\sqrt{-1}) are all complex, so there is no choice of “conductor” which differentiates between complex numbers with positive or negative real part at the infinite places in \mathbf{A}^{\times}_F.

Noah’s proof of the theorem above exploits the following idea. Potential modularity not only gives meromorphy of the L-function, but more importantly (in this case) holomorphy and non-vanishing in the (analytically normalized) halfplane Re(s) >= 1. Moreover, again using functorialities, potential automorphy, and results of Shahidi, one obtains similar results not only for the degree 4 L-function, but also the degree 5 L-function, and also crucially the Rankin-Selberg L-functions of degrees 16, 20, and 25. From this one can obtain various “prime number theorem” estimates for quantities involving the Frobenius eigenvalues, and then one has to massage these into an inequality. A simple version of this argument is as follows: given some infinite set of real numbers a_n \in [-2,2] such that

\displaystyle{\frac{1}{n} \sum_{i=1}^n a_i \rightarrow 0, \qquad \frac{1}{n} \sum_{i=1}^n a^2_i \rightarrow 1,}

One can draw the conclusion that a_n > 1/2 - \epsilon infinitely often, by (for example) considering the average of the quantity (2a_n - 1)(a_n + 2). Moreover, this is the best possible bound given these constraints.

Note that since the Sato-Tate conjecture is known in all other cases, one only has to consider cases (3) and (4), which behave slightly differently in this argument. In fact, in case (3), one can do much better:

Theorem (Noah Taylor). Let C be a curve over a totally real field F such that A = \mathrm{Jac}(C) is of type (3). Then there exists a positive density of primes \mathfrak{p} (with N(\mathfrak{p}) = p), such that

\displaystyle{\# C(\mathcal{O}/\mathfrak{p}) - p  - 1 >  2.47 \sqrt{p}}.

(Note that once this result is known in case (3) it is known for all curves whose Jacobian is not of type (4), that is, those whose Jacobians admit a non-trivial endomorphism over \mathbf{C}.) The point is that, in this case, one knows not just the potential automorphy of A, but also the potential automorphy of the corresponding two-dimensional representations over the quadratic extension L, and so one can also exploit the automorphy of symmetric powers of the corresponding GL(2)-automorphic representations (and further analyticity results for higher symmetric powers) as well as a zoo of Rankin-Selberg L-functions coming from pairs of low symmetric powers. (As for the constants involved in both of these theorems, they are essentially optimal given the automorphic input.)

These results tie in to problems raised in various talks of Nick Katz (see for example this talk). Noah’s result above implies that, given an curve C over a totally real field, one can tell that it doesn’t have genus one from the distribution of the traces of Frobenius except possibly in the case when its Jacobian has no non-trivial geometric endomorphism (the “typical” case, of course). It’s a little sad that the modularity results are not sufficient to handle that last case as well — showing that the support of the normalized trace of Frobenius extends beyond [-2,2] would require knowing something close to functoriality of the map \mathrm{Sym}^2: \mathrm{GL}(4) \rightarrow \mathrm{GL}(10), and this is currently out of reach, unfortunately. Oh well, that’s a shame: wow I dearly would have loved to give a talk entitled Simple things that Nick Katz doesn’t know (but I do).

Posted in Mathematics | Tagged , , , , , , , , , , , , , , , , , , , , , , , , , | 2 Comments

Cookies for everyone!

Chicago University has recently moved to two-factor authentication. While this is not itself objectionable, it is pretty annoying to have to have access my cell phone for push notifications just to be able to use MathSciNet. Fortunately, there is (at least) an option to “remember” a login for 30 days without having to use two-factor authentication. That is, it would be fortunate if this feature actually worked. Having just converted to Firefox from the interminably slow Safari, the uchicago website prevents me from remembering my log in for 30 days because “I don’t have cookies enabled.” Except I do have cookies enabled, just not from third parties. What does the crack security team at Chicago ITS suggest? Enable all third parties cookies from everyone! Firefox does have an option from allowing (third party) cookies from an approved list of websites, but, at this time, they couldn’t work out all the different websites that wanted force a cookie upon me when logging in.

 added: The final word from Chicago ITS:

Hello,

We have looked into other reasons as to why this functionality is the way it is and it seems to be a part of DUO authentication that cannot be changed. The best suggestion we [have is] using VPN when accessing university resources if you want it to remember you for 30 days. Otherwise, 3rd party cookies will have to be enabled to have this work without using VPN. Using the University VPN does have advantages if you are working in a public network.

Regards,

ITS Service Desk

====================

Posted in Rant | Tagged , , , | 13 Comments

Hear my prayer

Music post!

Current obsession:

This plaintive choral work, possibly an incomplete fragment of an unfinished anthem, is essentially one long phrase building to a final resolution with just the right amount of dissonance. There are a number of fine recordings on youtube, but this performance is my current favourite. (It appears to be tuned in something close to a baroque pitch with A = 415, but I’m neither sure how accurate that estimate is nor how deliberate that specific choice was — those of you who know how to take Fourier transforms in practice can tell me how close I am.)

Edit: OK, thanks to an old answer here by user Lenoil (I love it how the Stack Exchange websites are stocked with competent experts able to answer so many questions; a pity there is no equivalent in mathematics), I could at least clip out the very first note in the file and let Mathematica compute the corresponding Fourier transform. There’s still a little vibrato which makes for inaccuracies, but here was the result from the opening “hear”:

Frequency Graph

Assuming (very generously to myself) that the second peak is the result of vibrato, I think this suggests an opening note very close to 245Hz. The piece itself is in C (minor) and opens on a middle C. Assuming (equal temperament) an A = 415 tuning, the resulting middle C should in fact be approximately

415 \cdot 2^{-9/12} \sim 246.7

Make of that what you will (probably not so much, I imagine).

Posted in Music | Tagged , , , , , , , | 2 Comments

Mazur 80

Last week I was in Cambridge for Barry’s 80th birthday conference. If you are wondering why it took so long for Barry to get a birthday conference, that’s probably because you didn’t know that there was *also* a 60th birthday conference (in 1998, which is not entirely obvious given Barry’s actual birthday). It dates me somewhat to remark that I arrived at Berkeley as a graduate student just in time to miss this conference. (Come July 1, or thereabouts, I will have spent 20 years living in the US.) My first memory of Barry dates from when I was a graduate student in Berkeley. Ken introduced us; we chatted in the tea room (1015), and, as I remember, Barry listened and talked to me with much more generosity and patience than anything I had to say particularly warranted. My next interactions came about through my work with Kevin Buzzard on various conjectures relating to the Eigencurve. Once again, the level of enthusiasm he expressed for our ideas was the type of positive feedback that (to put it mildly) comes somewhat infrequently in academia. He agreed to be my NSF sponsor at Harvard, and later we became co-authors and friends. Needless to say, I was psyched about coming to this conference, and the conference was great! I will not rehash the talks here, beyond a few small observations.

Jordan was very careful about notation in his talk. He had previously used X as a symbol to denote an integer, so he carefully used S to denote an object which admits a map to B\mathbf{Z}:

Jordan

Answered here: What is BG? Not answered here: why do two corners of this pullback square look suspiciously like the same symbol?

As for my own talk, having previously tried to give some technical talks on math related to the CG-method which went horribly wrong, this time I gave a slide talk on my work with Boxer, Gee, and Pilloni which was all candy and no vegetables. (Summary of talk: first Riemann, then Wiles, and now us). Kai-Wen legitimately expressed disappointment at the lack of details (fair enough, you can’t fault that guy for skipping details). Otherwise, it elicited the reminder from Dick Gross that — although I could get by and make a living doing this sort of thing — it was time for number theorists to escape the “ghetto of holomorphic forms” (a phrase I think he attributed to someone else, I should say). Hey, Dick, don’t I at least get points from escaping the even worse “swamplands of discrete series”?

For those playing Barry Mazur bingo (sample squares: Gorenstein, “but…that’s beautiful”, X_0(11)) there were plenty of opportunities to see the influence of Barry’s mathematics. There was, however, a novel aspect of the conference which was an interdisciplinary day consisting of three conversation sessions of Literature/Poetry, History of Science, and Philosophy/Law/Physics respectively. By all accounts this was a wildly successful enterprise (hat-tip to the organizers). I did have one question I would have liked to ask one of the historians of mathematics, but the theme of the conversation meandered elsewhere. Instead I shall ask it here into the void (I’m not accusing you, dear reader, of being a void, merely that there are probably not any actual historians reading this blog):

A working mathematician usually has a very interpretative (and somewhat anachronistic) view of the history of mathematics: Galois “knew” which groups \mathrm{PSL}_2(\mathbf{F}_p) acted on p points, Gauss “knew” XYZ about class groups, etc. Mathematicians feel confident in these interpretations even if they are not explicitly written in this form in the original texts. What are the dangers in this (Whiggish?) view of the history of mathematics?

Cambridge Culinary Roundup:

With conference banquets (with some touching and amusing speeches by those who knew Barry well) and receptions going on, there was only a limited time for dining, not to mention the problem of trying to book restaurants at the last moment. Still, there was some opportunity to revisit some familiar and some new places:

Burger versus Burger: When it comes to Cambridge burgers, there is only one possible choice…or is there? My general impression was that the only way you could like Mr Bartley’s was if you were first exposed to it before your culinary tastes had a chance to develop (i.e. as a drunk undergraduate). On the other hand, a Cragie on Main burger (circa 2012) was as close in my mind to burger perfection as you could get. But did either of these opinions hold up today? Thus was the origin of the burger versus burger challenge. The participants for round one (Tuesday lunch) included myself, Quomodocumque,
The Hawk, Akshay, Joel, and Bisi. (Although Bisi was participating in a slightly different show, namely the latest episode of “mathematicians trick Bisi into going to a grungy restaurant.”) Round two was Tuesday dinner. Bisi and Joel dropped out on the reasonable basis that they had already consumed enough saturated fat for one week, but the rest of us continued on.

The conclusion? Cragie on Main clearly serves the superior burger (as noted by the Hawk, the fact that the request for a “medium rare” burger came out medium rare at Cragie on Main versus medium well at Bartley’s meant there could be no other conclusion). But perhaps inevitably, my opinions were forced to be somewhat softened in both directions. Bartley’s really did a decent job as far as the overall taste was concerned, and Cragie on Main’s burger — while better — stopped well short of being transformative. I suspect that they’ve been coasting for too long and haven’t maintained the level of excellence they started with (maybe that’s true of Bartley’s too, although I didn’t get a chance to eat there in 1960). In fact, there’s a generally sound principle to be a little wary about restaurants which have been around for too long. (Having said that, I would still find it hard to skip going to Rivoli restaurant on my next trip to Berkeley.)

Chess: Au Bon Pain has disappeared! The entire Holyoke center building is under some sort of reconstruction. The chess players are still around, however, having moved to (literally) the triangle that is Harvard square. I played a few games, and was prettily solidly crushed by a 2300 player in some lightning games. I also declined to play a $10 lightning game against an IM with generous odds, not because I thought I didn’t have a 50% chance of winning, but because I didn’t think I had a 90% chance of winning, and losing would have been at least 10 times more annoying than winning would have been pleasant.

Coffee: 5 (or so) years ago, Crema was a revolution in Harvard Square (i.e. drinkable coffee, reasonable hipster attitude). While their coffee was never at the level of something like Voltage Coffee (near MIT, and sadly now gone), it made staying at the car park known as the Harvard Square Hotel a more palatable option than at the “quaint” Irving street B&B. Times have changed! Crema is a victim of its own success — in a busy place which requires a frequent changeover of staff, the emphasis on coffee no longer seems to be paramount, and the quality control has dropped precipitously. The result was high inconsistency. Out of four coffees I got there, two were OK, one was pretty bad, and one was send directly into the bin. (I would like to have said “tossed in an elegant arc directly into the rubbish bin,” but if I had really attempted that, it would have been more like “unceremoniously spilt all over my shirt.”) As of today, there are definitely better options even in Harvard Square (further afield, one trip was made to Broadsheet which showed promise, even though my own flat white there was merely acceptable). Namely: Tatte Bakery & Cafe, which I really quite liked as far as the pastries and the sandwiches went, and the coffee itself rose to acceptable if not excellent standards.

Darwin’s is still Darwin’s (I prefer Tatte), Night Market (inspired by asian street food) was pretty interesting (some tasty eggplant) if a little idiosyncratic, and Parsnip did a perfectly good job of replacing “Upstairs at the Square,” a restaurant at which I had many a dinner when I used to live in Cambridge. (I had my eye on a few other restaurants, but none of them could take at short notice a reservation for 4 on a Thursday, so Parsnip was especially good given the constraint of not being so popular.

If you have suggestions of better coffee that I may have missed, please make suggestions since I will be returning in November. I also hope the Cambridge weather in November is more like June weather, given that the weather this week was more like November weather:

Cambridge Clouds

Posted in Mathematics | Tagged , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , | 16 Comments

Upcoming Attractions

There’s a packed schedule for graduate classes at Chicago next Fall: Ngô Bảo Châu on automorphic forms (TueTh 11:00-12:30), Akhil Mathew on perfectoid spaces (MWF 12:30-1:30), and George Boxer and me on (higher) Hida theory (MW 1:30-3:00). Strap yourself in!

Slightly more into the future and for a slightly different audience, it has now bene confirmed that there will be a special semester on “The arithmetic of the Langlands program” at the Hausdorff Institute in Bonn during May 4 — August 21 in 2020 (organized by Ana Caraiani, Laurent Fargues, Peter Scholze, and me).

Posted in Mathematics | Tagged , , , , , , , , , | 1 Comment

Nobody Cares About Your Paper

I handle quite a few papers (though far fewer than other editors I know) as an associate editor at Mathematische Zeitschrift. Since the acceptance rate at Math Zeit is something in the neighbourhood of 20%, there are certainly good papers which I have to reject. What papers “make the cut” through the first round depends, to some extent, on how “interesting” the paper is. Naturally, this is a somewhat subjective judgement. I make the determination (in part) on quick opinions I solicit from experts. But there is also a second possible mechanism available. Suppose I decide to send the paper out for a thorough review, but then I can’t find anyone to review it. If I email 10 people in the immediate field (not at the same time of course; usually requests to review come with a request for alternative suggestions for reviewers) and they all say no, does that indicate that nobody cares about your paper and it should be rejected? What if it’s twenty people? I haven’t (yet) ever rejected a paper on these grounds. But I have started to form opinions on certain specific subfields of number theory which seem to generate many pages of material but very few people willing to review anything. If they don’t care enough about their own subject to bother reviewing each other’s papers, why should anyone else?

Posted in Mathematics, Politics, Rant | Tagged , | 6 Comments