This Week’s Finds In Number Theory

Twenty years ago yesterday, John Baez posted the first installment of This Week’s Finds in Mathematical Physics.  In so doing, he invented the math blog, and, quite possibly, the blog itself.  A lot of mathematicians of my generation found in John’s blog an accessible, informal, but never dumbed-down window beyond what we were learning in classes, into the messy and contentious ground of current research.  And everybody who blogs now owes him a gigantic debt.

In his honor I thought it would be a good idea to post a “This Week’s Finds” style post of my own, with capsule summaries of a few papers I’ve recently noted with pleasure and interest.  I won’t be able to weave these into a story the way John often did, though!  Nor will there be awesome ASCII graphics.  Nor will any of the papers actually be from this week, because I’m a little behind on my math.NT abstract scanning.

If you run a math blog, please consider doing the same in your own field!  I’ll link to it.

Update:  It begins!  Valeria de Palva offers This Week’s Finds In Categorical Logic.  And Matt Ward, a grad student at UW-Seattle, has This Week’s Finds in Arithmetic Geometry.

1)  “On sets defining few ordinary lines,” by Ben Green and Terry Tao.

The idea that has launched a thousand papers in additive combinatorics:  if you are a set approximately closed under some kind of relation, then you are approximately a set which is actually closed under that kind of relation.  Subset of a group mostly closed under multiplication?  You must be close to an honest subgroup.  Subset of Z with too many pair-sums agreeing?  You have an unusually large intersection with an authentic arithmetic progression.  And so on.

This new paper considers the case of sets in R^2 with few ordinary lines; that is, sets S such that most lines that intersect S at all intersect S in three or more points.  How can you cook up a set of points with this property?  There are various boring ways, like making all the points collinear.  But there’s only one interesting way I can think of:  have the points form an “arithmetic progression” …,-3P,-2P, -P, P,2P,3P, …. in an elliptic curve!  (A finite subgroup also works.)  Then the usual description of the group law on the curve tells us that the line joining two points of S quite often passes through a third.  Green and Tao prove a remarkable quasi-converse to this fact:  if a set has few ordinary lines, it must be concentrated on a cubic algebraic curve!  This might be my favorite “approximately structured implies approximates a structure” theorem yet.

2) “Asymptotic behavior of rational curves,” by David Bourqui.  Oh, I was about to start writing this but when I searched I realized I already blogged about this paper when it came out!  I leave this here because the paper is just as interesting now as it was then…

3) “The fluctuations in the number of points of smooth plane curves over finite fields,” by Alina Bucur, Chantal David, Brooke Feigon, and Matilde Lalin;

“The probability that a complete intersection is smooth,” by Alina Bucur and Kiran Kedlaya;

“The distribution of the number of points on trigonal curves over F_q,” by Melanie Matchett Wood;

“Semiample Bertini theorems over finite fields,” by Daniel Erman and Melanie Matchett Wood.

How many rational points does a curve over F_q have?  We discussed this question here a few years ago, coming to no clear conclusion.  I still maintain that if the curve is understood to vary over M_g(F_q), with q fixed and g growing, the problem is ridiculously hard.

But in more manageable families of curves, we now know a lot more than we did in 2008.

You might guess, of course, that the average number of points should be q+1; if you have to reason to think of Frobenius as biased towards having positive or negative trace, why not guess that the trace, on average, is 0?  Bucur-David-Feigon-Lalin prove that this is exactly the case for a random smooth plane curve.  It’s not hard to check that this holds for a random hyperelliptic curve as well.  But for a random trigonal curve, Wood proves that the answer is different — the average is slightly less than q+2!

Where did the extra point come from?

Here’s one way I like to think of it.  This is very vague, and proves nothing, of course.  The trigonal curve X has a degree-3 map to P^1, which is ramified at some divisor D in P^1.  If D is a random divisor, it has one F_q-point on average.  How many F_q-points on X lie over each rational point P of D?  Well, generically, the ramification is going to be simple, and this means that there are two rational points over D; the branch point, and the unique unramified point.  Over every other F_q-point of D, the Frobenius action on the preimage in X should be a random element of S_3, with an average of one fixed point.  To sum up, in expectation we should see q rational points of X over q non-branch rational points of P^1, and 2 rational points of X over a single rational branch point in P^1, for a total of q+2.

(Erman and Wood, in a paper released just a few months ago, prove much more general results of a similar flavor about smooth members of linear systems on P^1 x P^1 (or other Hirzebruch surfaces, or other varieties entirely) which are semiample; for instance, they may have a map to P^1 which stays constant in degree, while their intersection with another divisor gets larger and larger.)

Most mysterious of all is the theorem of Bucur and Kedlaya, which shows (among other things) that if X is a random smooth intersection of two hypersurfaces of large degree in P^3, then the size of |X(F_q)| is slightly less than q+1 on average.  For this phenomenon I have no heuristic explanation at all.  What’s keeping the points away?

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16 thoughts on “This Week’s Finds In Number Theory

  1. NDE says:

    Is there a similar result for coplanar quadruples in Z^3, and likewise in higher dimension? One might guess it’s a normal elliptic curve or some degeneration thereof.

  2. Tom Leinster says:

    You’re missing a word in the title of Green and Tao’s paper.

  3. Valeria de Paiva says:

    Hi, I liked the idea of celebrating 20 years of “This week’s finds in Mathematical Physics” with similar posts and tried to come up with my own “This week’s finds in Categorical Logic”
    http://logic-forall.blogspot.com/2013/01/this-week-in-categorical-logic-baez.html. Thanks for the suggestion!

  4. Jason Starr says:

    Regarding Bourqui’s motivic Batyrev-Manin, do you know if there is a motivic analogue of the height bounds implied by Batyrev-Manin? As I understand it, Batyrev-Manin roughly predicts existence of rational points of height bounded by 1 over the Tamagawa number. Is there an analogue of this in Bourqui’s work?

  5. Jason Starr says:

    I looked at Bourqui’s paper and I see now that he is looking at morphisms from the projective line to a fixed Fano, i.e., sections of a constant family. So there are always constant sections, i.e., height 0.

  6. JSE says:

    Right — and in that spirit, extending his stuff to spaces of sections of a family of Fanos fibered over P^1 is a very natural project.

  7. Terence Tao says:

    @JSE: Thanks for the shout-out!

    @NDE: I don’t know of any counterexamples to such a higher-dimensional statement, but unfortunately our argument is extremely two-dimensional in nature (relying in particular and quite crucially on Euler’s formula V-E+F=2 applied to the projective dual of the configuration of points and lines). Actually, because of our use of Euler’s formula, our argument is also extremely real in nature; a similar result should hold for points in C^2 rather than R^2, but the first step of our arguments stops working (though, tantalisingly enough, the remainder of the arguments still look mostly viable). My guess though is that there should be an alternate proof of these results which extends more easily to higher dimensions.

  8. Tom Leinster says:

    Joachim Kock has written a piece too (in the ASCII style of the old This Week’s Findses): String diagrams, the number 5, and the moons of Jupiter.

  9. [...] I should mention other tributes to TWF here, here, and [...]

  10. As for “what’s keeping the points away” in P^3–the answer is nothing, that’s why there is slightly less than q+1 but only slightly. The style of argument you make for why there is an extra point for trigonal curves is only keeping track of “whole points,” e.g. when you talk about generically the ramification is going to be simple, if you kept track of the error that could introduce, it is o(1). And Bucur and Kedlaya get q+1+o(1) points on complete intersection curves in P^3. So at the level of detail of the conversation that is basically q+1.

    In fact, one can apply the heuristic you give more precisely and obtain exactly the answer I obtain in my trigonal curves paper. And applying “the same” heuristic for P^3 will give you exactly the B-K answer, under a suitable description of the heuristic. Here’s my description: You have map curve –> X for some fixed X, look at what can happen locally above rational points of X, and then assume the fibers are independent. In the case of trigonal curves, X=P^1 and the map is a triple cover, but in the case of complete intersections in P^3, we have X=P^3 and the map is an embedding.

    In my paper with Erman, what our main result allows one to do is to take heuristic arguments as I just described (and in fact “curve” can be replaced with other things as well) and turn them into theorems–we precisely prove that what goes on in the fibers is independent.

  11. JSE says:

    Good answer!

    I guess another way to think of this is: the “extra point” on the trigonal curves says that if X is the universal trigonal curve over the moduli of trigonal curves, then X(F_q) is (q+2)*number of trigonal curves instead of (q+1)*number of trigonal curves, up to smaller factors; so this is a signal that there’s an extra divisor on X, which presumably is something like the branch locus in the universal trigonal curve.

    So getting q+1-[something on order 1/q] maybe says that there’s no “surprising” divisors on the universal curve X over the Hilbert scheme of space curves, but there might be some kind of class in, I guess, H^3? which is making a negative contribution to X(F_q) on order q^{-2} |X(F_q)|.

  12. [...] Baez’s This Week’s Finds vary so much, I’ll just copy what Jordan Ellenberg did here and give some papers posted the last week that caught my [...]

  13. hilbertthm90 says:

    You can tack my name on if you want: Matt Ward. I tried to remain anonymous at the beginning of my blogging, but a few years ago I put my name next to my blog wherever I found it listed (upon searching now, I actually can’t find those lists anymore).

  14. I actually think of the “extra” divisor on the universal trigonal curve as being the one associated to the line bundle that is just pulled back from the map to P^1. For hyperelliptic curves, it would just be the canonical bundle so it isn’t “extra.” In my trigonal curves paper, I give the counts that would result for n-gonal curves for any n from a function field version of Bhargava’s heuristics for counting S_n number fields–one predicts average q+2+o(1) for n-gonal curves over F_q for any fixed n>2. I think of the extra point as coming from that same extra bundle.

  15. [...] members, both joining us next fall:  Daniel Erman in commutative algebra and algebraic geometry (seen previously on the blog counting smooth members in semiample linear systems over finite fields) and Uri Andrews in model [...]

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