Bounded rank was probable in 1950

Somehow I wrote that last post about bounded ranks without knowing about this paper by Mark Watkins and many other authors, which studies in great detail the variation in ranks in quadratic twists of the congruent number curve.  I’ll no doubt have more to say about this later, but I just wanted to remark on a footnote; they say they learned from Fernando Rodriguez-Villegas that Neron wrote in 1950:

On ignore s’il existe pour toutes les cubiques rationnelles, appartenant a un corps donné une borne absolute du rang. L’existence de cette borne est cependant considérée comme probable.

So when I said the conventional wisdom is shifting from “unbounded rank” towards “bounded rank,” I didn’t tell the whole story — maybe the conventional wisdom started at “bounded rank” and is now shifting back!

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Show your work

Here’s another comment on that New York Times piece:

“mystery number game …. ‘I’m thinking of a mystery number, and when I multiply it by 2 and add 7, I get 29; what’s the mystery number?’ “

See, that’s what I mean, the ubiquitous Common Core approach to math teaching these days wouldn’t allow for either “games” or “mystery”: they would insist that your son provide a descriptive narrative of his thought process that explains how he got his answer, they would insist on him drawing some matrix or diagram to show who that process is represented pictorially.

And your son would be graded on his ability to provide this narrative and draw this diagram of his thought process, not on his ability to get the right answer (which in child prodigies and genius, by definition, is out of the ordinary, probably indescribable).

Actually, I do often ask CJ to talk out his process after we do a mystery number.  I share with the commenter the worry of slipping into a classroom regime where students are graded on their ability to recite the “correct” process.  But in general, I think asking about process is great.  For one thing, I learn a lot about how arithmetic facility develops in the mind.  I asked CJ the other night how many candies he could buy if each one cost 7 cents and he had a dollar.  He got the right answer, 14, not instantly but after a little thought.  I asked him how he got 14 and he said, “Three 7s is 21, and five 21s is a dollar and five cents, so 15 candies is a little too much, so it must be 14.”

How would you have done it?

“Like a girl”

I wrote a New York Times op/ed last week about the relationship between teaching math and coaching Little League.  Several people wrote me to say that I shouldn’t have written the following passage:

My level of skill at baseball — actually, with every kind of ball — is pretty much the opposite of my mastery of math. I’ve reached 40 and I still throw in the way that we used to call, before they started showing college softball on TV, “like a girl.”

So obviously my goal here is to undercut the stereotype and present it as obsolete.  But the people who wrote me argued that to use the force of a sexist phrase to give my sentence a little oomph is a problem, even if (as I once heard J. P. Serre say about a piece of notation) “I mention it only in order to object to it.”

So I asked about this on Facebook, and maybe 60% of people thought it was fine, and 40% said that they winced when they read it.

Which means it’s not fine.  Because why write something that makes 40% of readers wince in annoyance?  Especially when a) it’s in no way intrinsic to the piece, which is otherwise not about gender roles, and b) the piece itself ties math to baseball, a boy-coded activity, and has much more material about my son than it does about my daughter.

I think “like a girl” can be an OK place to go if you need to.  But I didn’t need to.  So I think I shouldn’t have.

One of my friends suggested I should have said instead that I throw “like a mathematician.”  Better!


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R.E.M. live at the Rockpalast, 2 Oct 1985

Complete show on YouTube.  In case you were wondering what the fuss was about.

Rank 2 versus rank 3

One interesting feature of the heuristics of Garton, Park, Poonen, Wood, Voight, discussed here previously: they predict there are fewer elliptic curves of rank 3 than there are of rank 2.  Is this what we believe?  On one hand, you might believe that having three independent points should be “harder” than having only two.  But there’s the parity issue.  All right-thinking people believe that there are equally many rank 0 and rank 1 elliptic curves, because 100% of curves with even parity have rank 0, and 100% of curves with odd parity have rank 1.  If a curve has even parity, all that has to happen to force it to have rank 2 is to have a non-torsion point.  And if a curve has odd parity, all that has to happen to force it to have rank 3 is to have one more non-torsion point you don’t know about it.  So in that sense, it seems “equally hard” to have rank 2 or rank 3, given that parity should be even half the time and odd half the time.

So my intuition about this question is very weak.  What’s yours?  Should rank 3 be less common than rank 2?  The same?  More common?


For some reason I was thinking about pieces of culture that have departed from the world but which somehow didn’t “stick” well enough to persist even in the sphere of nostalgia.  Like when people think about the early 1990s, the years when I was in college, they might well say “oh yeah, grunge” or “oh yeah, wearing used gas station T-shirts with a name stitched on” or “oh yeah, Twin Peaks” or “oh yeah, OK Soda” or whatever.

But no one says “oh yeah, Fido Dido.”  So here I am doing it.

It is inherently hard to try to list things you’ve forgotten about.  My list right now consists of

  • Fido Dido
  • Saying “bite me”
  • Smartfood
  • Devil sticks (from Jason Starr)

That’s it.  What have you got?

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Cool song, bro

I was in Barriques and “Bra,” by Cymande came on, and I was like, cool song, cool of Barriques to be playing this song that I’m cool for knowing about, maybe I should go say something to show everyone that I already know this cool song, and then I thought, why do I know about this song anyway? and I remembered that it was because sometime last year it was playing in Barriques and I was like, what is this song, it’s cool? and I Shazammed it.

So I guess what I’m saying is, I’m probably going to the right coffee shop.  Also, this song is cool.  I’m sort of fascinated by the long instrumental break that starts around 2:50.  It doesn’t seem like very much is happening; why is it so captivating?  I think my confusion on this point has something to do with my lack of understanding of drums.

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How do you share your New York Times?

My op/ed about math teaching and Little League coaching is the most emailed article in the New York Times today.  Very cool!

But here’s something interesting; it’s only the 14th most viewed article, the 6th most tweeted, and the 6th most shared on Facebook.  On the other hand, this article about child refugees from Honduras is

#14 most emailed

#1 most viewed

#1 most shared on Facebook

#1 most tweeted

while Paul Krugman’s column about California is

#4 most emailed

#3 most viewed

#4 most shared on Facebook

#7 most tweeted.

Why are some articles, like mine, much more emailed than tweeted, while others, like the one about refugees, much more tweeted than emailed, and others still, like Krugman’s, come out about even?  Is it always the case that views track tweets, not emails?  Not necessarily; an article about the commercial success and legal woes of conservative poo-stirrer Dinesh D’Souza is #3 most viewed, but only #13 in tweets (and #9 in emails.)  Today’s Gaza story has lots of tweets and views but not so many emails, like the Honduras piece, so maybe this is a pattern for international news?  Presumably people inside newspapers actually study stuff like this; is any of that research public?  Now I’m curious.



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Notes on Gone Girl

It reminds me of Martin Amis’s The Information, in that it is a really well-made thing, but one which I think probably shouldn’t have been made, and which I’m probably sorry I read, because it’s sick in its heart.

Everything else I can say is a spoiler so I’ll put it below a tab.

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Are ranks bounded?

Important update, 23 Jul:  I missed one very important thing about Bjorn’s talk:  it was about joint work with a bunch of other people, including one of my own former Ph.D. students, whom I left out of the original post!  Serious apologies.  I have modified the post to include everyone and make it clear that Bjorn was talking about a multiperson project.  There are also some inaccuracies in my second-hand description of the mathematics, which I will probably deal with by writing a new post later rather than fixing this one.

I was only able to get to two days of the arithmetic statistics workshop in Montreal, but it was really enjoyable!  And a pleasure to see that so many strong students are interested in working on this family of problems.

I arrived to late to hear Bjorn Poonen’s talk, where he made kind of a splash talking about joint work by Derek Garton, Jennifer Park, John Voight, Melanie Matchett Wood, and himself, offering some heuristic evidence that the Mordell-Weil ranks of elliptic curves over Q are bounded above.  I remember Andrew Granville suggesting eight or nine years ago that this might be the case.  At the time, it was an idea so far from conventional wisdom that it came across as a bit cheeky!  (Or maybe that’s just because Andrew often comes across as a bit cheeky…)

Why did we think there were elliptic curves of arbitrarily large rank over Q?  I suppose because we knew of no reason there shouldn’t be.  Is that a good reason?  It might be instructive to compare with the question of bounds for rational points on genus 2 curves.  We know by Faltings that |X(Q)| is finite for any genus 2 curve X, just as we know by Mordell-Weil that the rank of E(Q) is finite for any elliptic curve E.  But is there some absolute upper bound for |X(Q)|?  When I was in grad school, Lucia Caporaso, Joe Harris, and Barry Mazur proved a remarkable theorem:  that if Lang’s conjecture were true, there was some constant B such that |X(Q)| was at most B for every genus 2 curve X.  (And the same for any value of 2…)

Did this make people feel like |X(Q)| was uniformly bounded?  No!  That was considered ridiculous!  The Caporaso-Harris-Mazur theorem was thought of as evidence against Lang’s conjecture.  The three authors went around Harvard telling all the grad students about the theorem, saying — you guys are smart, go construct sequences of genus 2 curves with growing numbers of points, and boom, you’ve disproved Lang’s conjecture!

But none of us could.

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