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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Fan Mail

Edit: A previous version of this post has been edited upon request.

Somewhat less salaciously, I received a Christmas card from an academic couple of whom I am absolutely sure I have never met. (I just checked my mailbox at work for the first time this year, which is where this card was sent.) They work in a state I have never visited, and neither of them are mathematicians (though one appears to have a math PhD). They wished me the best “on my career.” Apparently (according to their website) one of them is an expert on “targeted killings.” I hope that “on my career” is not a euphemism.

As for mail more directly relevant to me and this blog, I did (three or so years ago?) receive an unsolicited package in the mail from a blog reader. The sender’s name (Leslie, I think?) was suitably unisex, so I naturally assumed that it was a swooning 20-something female who had fallen for my prose and occasional deliberate grammatical and spelling errors. But the reality was better: it was (as far as I could tell from a google search) a 60 year old male with a PhD in math, who send me a CD with some Schumann Lieder, in particular an Edith Mathis CD (with Christoph Eschenbach on the piano) entitled “Frauenliebe und Leben & other Lieder”. Absolutely wonderful! Through a quirk of fate this CD has ended up in my car, and has been in heavy rotation over the past year during my commute. Because I wouldn’t want you to miss out, I’ve given a youtube link to one of the songs below (Kennst du das Land — not from the titular cycle, but chosen in part because the accompaniment reminds me stylistically of Dichterliebe, partly because it sounds good, and partly because Robert wrote it for Clara and it is Valentine’s day). But it makes me wonder: what type of fan mail does Quomodocumque get? (or, for that matter, Terry Tao and Sir Timothy Gowers, FRS)

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Alinea

Fresh off my dining experiences in Jerusalem, I returned home for some more local dining.

From smoked juniper bushes to edible stones, I arrived at Alinea fearing that I would be underwhelmed but left leaving very happy. Rather than try to describe my meal in any detail, let me instead just post a few teaser pictures. I came into the experience without any preconceived notions of what to expect, which I recommend. (I also wasn’t paying, which I also recommend.)

Stones

The trick is to guess how many of these are edible.

Juniper

The decorations and the food are intertwined. This was a feature of many dishes of this —and I am guessing most — meals at Alinea.

Berries

This had some Foie Gras, I believe.

Dessert Stones

This chocolate mousse hiding in one of these stones was delicious.

Balloon

I am not a number! I am a free man!

Rover

Many dishes omitted, of course. One notable course involved a pomegranate cocktail which had a slight bouquet of christmas pudding; possibly the best cocktail I have ever had. Indeed, that has inspired one of my upcoming culinary choices for next two weeks, which include: the Violet Hour, L’Etoile for the number theory seminar dinner at Madison (this one is more of a suggestion than a concrete plan, but JSE, can we make it happen?), and the Victoria Market.

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