Tag Archives: melanie matchett wood

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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Some visitors, and countable unions

A busy few days: we had a run of interesting visitors this week in Madison, including Thomas Lam, who gave a beautiful talk about total positivity (a subfield of algebraic combinatorics, not a self-help philosophy); Melanie Matchett Wood, who explained how to parametrize binary forms of degree n in the Bhargava style, not only over Z but over an arbitrary base scheme (which is to say, not really in the Bhargava style!); and Davesh Maulik, who showed us how one can rather miraculously count rational curves on a single K3 by counting rational curves on a suitably chosen one-parameter family of K3s, and then “dividing” by the Noether-Lefschetz theory attached to the family. Very agreeably for number theorists, a key point is the product formulae of Borcherds, which provide modular forms on the moduli space of K3s whose zeroes and poles are supported on the countable union of subvarieties where the Picard number jumps upwards from its generic value.

This led to an amusing conversation at lunch about countable unions of subvarieties. Here’s a remark: if A is an abelian variety over the complex numbers, it’s completely obvious that A(C) contains some non-torsion points; the torsion locus is a countable union of varieties of strictly lower dimension (in this case 0) and thus can’t cover A(C). On the other hand, if A is over Fpbar, every point of A(Fpbar) is defined over some finite field, and thus all these points are torsion. The case of Qbar is intermediate in difficulty; indeed, there are nontorsion points on every abelian variety over Qbar, but this is not, in some sense, by “pure thought” — one might, for instance, use the argument that torsion points have height 0 but that there are plainly points of arbitrarily large height on A(Qbar). This uses some actual theorems, not just a comparison of cardinalities. Similarly, one can ask: are there elliptic curves over an algebraically closed field k with End(E/k) = Z? When k = C, the answer is obviously yes. When k = Fpbar, the answer is no, thanks to Frobenius. And when k is Qbar, the answer is again no, but maybe one has to use a bit more — for instance, that a CM elliptic curve over a number field has potentially good reduction everywhere.

In general, it’s pretty hard to see whether a countable union of subvarieties of X/Qbar covers all the Qbar-points! Here are two well-known open questions in this vein.

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