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).

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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

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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).

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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?

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Chicago Seminar Roundup

Here are two questions I had about the past two number theory seminars. I haven’t had the opportunity to think about either of them seriously, so they may be easy (or more likely stupid).

Anthony Várilly-Alvarado: Honestly, I’ve never quite forgiven this guy for his behavior as an undergraduate. He was my TA when I taught Complex Analysis at Harvard, and he had the bad manners to do an absolutely wonderful job and be beloved by all the students. Nothing makes a (first time) professor look worse than a good TA. (It means I can’t even take any credit for the students in the class who became research mathematicians). Anyway, Tony gave a talk on his joint work with Dan Abramovich about the relation between Vojta’s conjecture and the problem of uniform bounds on torsion for abelian varieties. (Spoiler: one implies the other.) More specifically, assuming Vojta’s conjecture, there a universal bound on m (depending only on g and K) beyond which no abelian variety of dimension g over K can have full level structure.

If one wanted to prove this (say) for elliptic curves, and one was allowed to use any conjecture you pleased, you could do the following. Assume that E[m] = \mu_m \oplus \mathbf{Z}/m \mathbf{Z} for some large integer m. One first observes (by Neron-Ogg-Shafarevich plus epsilon) that E has to have semi-stable reduction at primes dividing N_E. Then the discriminant \Delta must be an mth power, and then Szpiro’s Conjecture (which is the same as the ABC conjecture) implies the desired result.

If you try to do the same thing in higher dimensions, you similarly deduce that A must have semi-stable reduction at primes dividing N_E. edit: some nonsense removed. One then gets implications on the structure of the Neron model at these bad primes, which one can hope to parlay in order to get information about local quantities associated to A analogous to the discriminant being a perfect power. But I’m not sure what generalizations of Szpiro’s conjecture there are to abelian varieties. A quick search found one formulation attributed to Hindry in terms of Faltings height, but it was not immediately apparent if one could directly deduce the desired result from this conjecture, nor what the relationship was with these generalizations to either ABC or to Vojta’s conjecture.

Ilya Khayutin: Ilya mentioned Linnik’s theorem that, if one ranges over imaginary quadratic fields in which a fixed small prime is split, the CM j-invariants become equidistributed. The role of the one fixed prime is to allow one to use ergodic methods relative to this prime. My naive question during the talk: given p is split, let \mathfrak{p} be a prime above p. Now one can take the subgroup of the class group corresponding to the powers of \mathfrak{p}. Do these equidistribute? The speaker’s response was along the lines that it would probably be quite easy to see this is false, but I didn’t have time after the talk to follow up. It’s certainly the case that, most of the time, the prime \mathfrak{p} will itself generate a subgroup of small index in the class group (the quotient will look like the random class group of a real quadratic field), but sometimes it will be quite large. For example, I guess one can take

\displaystyle{D = 2^n - 1, \qquad \mathfrak{p}^{n-2} = \left(\frac{1 + \sqrt{-D}}{2}\right)},

and the subgroup generated by this prime has order \log(D) compared to D^{1/2 + \epsilon}. So I decided (well, after writing this line in the blog I decided) to draw a picture for some choice of Mersenne prime. And then, after thinking a little how to draw the picture, realized it was unnecessary. The powers of \frak{p} in this case are given explicitly by

\displaystyle{\mathfrak{p}^m = \left(2^m, \frac{1 + \sqrt{-D}}{2}\right)},

It is transparent that for the first half of these classes, the first factor is much smaller than the second, but since the second term also has small real part, the ratio already lies inside the (standard) fundamental domain. Hence the corresponding points will lie far into the cusp. Similarly, the second half of the classes are just the inverses in the class group of the first half, and so will consist of the reflections of those points in x = 0 and so also be far into the cusp. So I guess the answer to my question is, indeed, a trivial no. So here is a second challenge: suppose that 2 AND 3 both split. Then do the CM points generated by \mathfrak{p} for primes above 2 AND 3 equidistribute? Actually, in this case, it’s not clear off the top of my head that one can easily write down discriminants for which the index of this group is large. But even if you can, sometimes \mathbf{Z}^2 subgroups get you much closer to equidistribution than \mathbf{Z}!

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The boundaries of Sato-Tate, part I

A caveat: the following questions are so obvious that they have surely been asked elsewhere, and possibly given much more convincing answers. References welcome!

The Sato-Tate conjecture implies that the normalized trace of Frobenius b_p \in [-2,2] for a non-CM elliptic curve is equidistributed with respect to the pushforward of the Haar measure of SU(2) under the trace map. This gives a perfectly good account of the behavior of the unnormalized a_p \in [-2 \sqrt{p},2 \sqrt{p}] over regions which have positive measure, namely, intervals of the form [r \sqrt{p},s \sqrt{p}] for distinct multiples of \sqrt{p}.

If one tries to make global conjectures on a finer scale, however, one quickly runs into difficult conjectures of Lang-Trotter type. For example, given a non-CM elliptic curve E over \mathbf{Q}, if you want to count the number of primes p < X such that a_p = 1 (say), an extremely generous interpretation of Sato-Tate would suggest that probability that a_p = 1 would be

\displaystyle{\frac{1}{4 \pi \sqrt{p}}},

and hence the number of such primes < X should be something like:

\displaystyle{\frac{X^{1/2}}{2 \pi \log(X)}},

except one also has to account for the fact that there are congruence obstructions/issues, so one should multiply this factor by a (possibly zero) constant depending one adelic image of the Galois representation. So maybe this does give something like Lang-Trotter.

But what happens at the other extreme end of the scale? Around the boundaries of the interval [-2,2], the Sato-Tate measure converges to zero with exponent one half. There is a trivial bound a_p  \le   t where t^2 is the largest square less than 4p. How often does one have an equality a^2_p = t^2? Again, being very rough and ready, the generous conjecture would suggest that this happens with probability very roughly equal to

\displaystyle{\frac{1}{6  \pi p^{3/4}}},

and hence the number of such primes < X should be something like:

\displaystyle{\frac{2 X^{1/4}}{3  \pi \log(X)}}.

Is it at all reasonable to expect X^{1/4 \pm \epsilon} primes of this form? If one takes the elliptic curve X_0(11), one finds a^2_p to be as big as possible for the following primes:

a_{2} = -2 \ge  -2 \sqrt{2} = -2.828\ldots,

a_{239} = -30 > -2 \sqrt{239} =  -30.919\ldots,

a_{6127019} = 4950   \le  2 \sqrt{p} = 4950.563\ldots,

but no more from the first 500,000 primes. That's not completely out of line for the formula above!

Possibly a more sensible thing to do is to simply ignore the Sato-Tate measure completely, and model E/\mathbf{F}_p by simply choosing a randomly chosen elliptic curve over \mathbf{F}_p. Now one can ask in this setting for the probability that a_p is as large as possible. Very roughly, the number of elliptic curves modulo p up to isomorphism is of order p, and the number with a_p = t is going to be approximately the class number of \mathbf{Q}(\sqrt{-D}) where -D = t^2 - 4p; perhaps it is even exactly equal to the class number H(t^2 - 4p) for some appropriate definition of the class number. Now the behaviour of this quantity is going to depend on how close 4p is to a square. If 4p is very slightly — say O(1) — more than a square, then H(t^2 - 4p) is pretty much a constant, and the expected probability going to be around 1 in p. On the other hand, for a generic value of p, the smallest value of t^2 - 4p will have order p^{1/2}, and then the class group will have approximate size p^{1/4 \pm \epsilon}, and so one (more or less) ends up with a heuristic fairly close to the prediction above (at least in the sense of the main term being around X^{1/4 \pm \epsilon}).

But why stop there? Let's push things even closer to the boundary. How small can a^2_p - 4p get relative to p? For example, let us restrict to the set S(\eta) of prime numbers p such that

\displaystyle{S(\eta):= \left\{p \ \left| \ p \in (n^2,n^2 + n^{2 \eta}) \ \text{for some} \ n \in \mathbf{Z} \right.\right\}}.

For such primes, the relative probability that a_p = \lfloor \sqrt{4p} \rfloor = 2n is approximately n^{\eta}/p \sim n^{2 \eta - 1}. So the expected number of primes with this property will be infinite providing that

\displaystyle{\sum \frac{n^{3 \eta}}{n^2 \log(n)}}

is infinite, or, in other words, when \eta \ge 1/3. So this leads to the following guess (don't call it a conjecture!):

Guess: Let E/\mathbf{Q} be an elliptic curve without CM. Is

\displaystyle{\liminf \frac{\log(a^2_p - 4p)}{\log(p)} = \frac{1}{3}?}

Of course, one can go crazy with even more outrageous guesses, but let me stop here before saying anything more stupid.

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Australiana

Some short observations from my recent trip:

Only in the same sense as Captain Renault could you possibly be shocked (shocked!) by what Bancroft drops into his pants.

The 90th percentile quality coffee in Melbourne (random mall coffee) is at (approximately) the level of the 10th percentile coffee in Chicago. While there’s plenty of good coffee in Chicago, you don’t want into a random cafe and expect to get something drinkable. You also don’t expect any random place to have a top of the line Marzocco machines. But if you want a few recommendations in the neighbourhood of either Lygon street or near the state library, I can suggest Market Lane/Pool House/Seven Seeds/Vincent the Dog/The League of Honest Coffee/Vertue of the Coffee Drink to get you started. Expert tip at US hipster cafes: order a magic (3/4 flat white with double ristretto), then look unimpressed when they don’t know what you are talking about.

More Coffee!More Coffee!Even More Coffee!

While you’re near the state library, stop off in the reading room for some speed chess (victory is mine!)

Australia has a lot of long beaches, and I don’t mean long in the sense of fractal dimension greater than one. I mean in the sense of having several miles of pristine beach to yourself:

Beach

Fight terrorism with philosophy! (and concrete bollards):

Chicago Philosophy

I always assumed that A’Beckett St was named after the turbulent priest. Not So! Apparently it is named after the first chief justice of Victoria. Upon learning this, I checked out the origins of the other street names in Melbourne’s CBD. Four of the North-South (ish) streets in order include (at some point) King-William-Queen-Elizabeth, and it is “common knowledge” that these streets are so named in pairs. Also false! William is named for King William IV, and Queen for Queen Adelaide, but King is named for Philip Gidley King, the governor of NSW from 1800-1806, and Elizabeth was “possibly” named for the wife of another Governor of NSW, Richard Bourke. (I did of course know that Bourke St (named after the guv) was not named after Burke, the explorer who (with Wills) became famous for his ludicrous incompetence.

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