## Soumya Sankar: Proportion of ordinarity in families of curves over finite fields

What’s the chance that a random curve has ordinary Jacobian? You might instinctively say “It must be probability 1” because the non-ordinary locus is a proper closed subvariety of M_g. (This is not obvious by pure thought, at least to me, and I don’t know who first proved it! I imagine you can check it by explicitly exhibiting a curve of each genus with ordinary Jacobian, but I’m not sure this is the best way.)

Anyway, the point is, this instinctive response is wrong! At least it’s wrong if you interpret the question the way I have in mind, which is to ask: given a random curve X of genus g over F_q, with g growing as q stays fixed, is there a limiting probability that X has ordinary Jacobian? And this might not be 1, in the same way that the probability that a random polynomial over F_q is squarefree is not 1, but 1-1/q.

Bryden Cais, David Zureick-Brown and I worked out some heuristic guesses for this problem several years ago, based on the idea that the Dieudonne module for a random curve might be a random Dieudonne module, and then working out in some detail what in the Sam Hill one might mean by “random Dieudonne module.” Then we did some numerical experiments which showed that our heuristic looked basically OK for plane curves of high degree, but pretty badly wrong for hyperelliptic curves of high genus. But there was no family of curves for which one could prove either that our heuristic was right or that it was wrong.

Now there is, thanks to my Ph.D. student Soumya Sankar. Unfortunately, there are still no families of curves for which our heuristics are provably right. But there are now several for which it is provably wrong!

15.7% of Artin-Schreier curves over F_2 (that is: Z/2Z-covers of P^1/F_2) are ordinary. (The heuristic proportion given in my paper with Cais and DZB is about 42%, which matches data drawn from plane curves reasonably well.) The reason Sankar can prove this is because, for Artin-Schreier curves, you can test ordinarity (or, more generally, compute the a-number) in terms of the numerical invariants of the ramification points; the a-number doesn’t care where the ramification points are, which would be a more difficult question.

On the other hand, 0% of Artin-Schreier curves over F are ordinary for any finite field of odd characteristic! What’s going on? It turns out that it’s only in characteristic 2 that the Artin-Schreier locus is irreducible; in larger characteristics, it turns out that the locus has irreducible components whose number grows with genus, and the ordinary curves live on only one of these components. This “explains” the rarity of ordinarity (though this fact alone doesn’t prove that the proportion of ordinarity goes to 0; Sankar does that another way.) Natural question: if you just look at the ordinary component, does the proportion of ordinary curves approach a limit? Sankar shows this proportion is bounded away from 0 in characteristic 3, but in larger characteristics the combinatorics get complicated! (All this stuff, you won’t be surprised to hear, relies on Rachel Pries’s work on the interaction of special loci in M_g with the Newton stratification.)

Sankar also treats the case of superelliptic curves y^n = f(x) in characteristic 2, which turns out to be like that of Artin-Schreier in odd characteristics; a lot of components, only one with ordinary points, probability of ordinarity going to zero.

Really nice paper which raises lots of questions! What about more refined invariants, like the shape of the Newton polygon? What about other families of curves? I’d be particularly interested to know what happens with trigonal curves which (at least in characteristic not 2 or 3, and maybe even then) feel more “generic” to me than curves with extra endomorphisms. Is there any hope for our poor suffering heuristics in a family like that?

## Random Dieudonne modules, random p-divisible groups, and random curves over finite fields

Bryden Cais, David Zureick-Brown and I have just posted a new paper,  “Random Dieudonne modules, random p-divisible groups, and random curves over finite fields.”

What’s the main idea?  It actually arose from a question David Bryden asked during Derek Garton‘s speciality exam.  We know by now that there is some insight to be gained about studying p-parts of class groups of number fields (the Cohen-Lenstra problem) by thinking about the analogous problem of studying class groups of function fields over F_l, where F_l has characteristic prime to p.

The question David asked was:  well, what about the p-part of the class group of a function field whose characteristic is equal to p?

That’s a different matter altogether.  The p-divisible group attached to the Jacobian of a curve C in characteristic l doesn’t contain very much information;  more or less it’s just a generalized symplectic matrix of rank 2g(C), defined up to conjugacy, and the Cohen-Lenstra heuristics ask this matrix to behave like a random matrix with respect to various natural statistics.

But p-divisible groups in characteristic p are where the fun is!  For instance, you can ask:

What is the probability that a random curve (resp. random hyperelliptic curve, resp. random plane curve, resp. random abelian variety) over F_q is ordinary?

In my view it’s sort of weird that nobody has asked this before!  But as far as I’ve been able to tell, this is the first time the question has been considered.

We generate lots of data, some of which is very illustrative and some of which is (to us) mysterious.  But data alone is not that useful — much better to have a heuristic model with which we can compare the data.  Setting up such a model is the main task of the paper.  Just as a p-divisible group in characteristic l is decribed by a matrix, a p-divisible group in characteristic p is described by its Dieudonné module;  this is just another linear-algebraic gadget, albeit a little more complicated than a matrix.  But it turns out there is a natural “uniform distribution” on isomorphism classes of  Dieudonné modules; we define this, work out its properties, and see what it would say about curves if indeed their Dieudonné modules were “random” in the sense of being drawn from this distribution.

To some extent, the resulting heuristics agree with data.  But in other cases, they don’t.  For instance:  the probability that a hyperelliptic curve of large genus over F_3 is ordinary appears in practice to be very close to 2/3.  But the probability that a smooth plane curve of large genus over F_3 is ordinary seems to be converging to the probability that a random Dieudonné module over F_3 is ordinary, which is

(1-1/3)(1-1/3^3)(1-1/3^5)….. = 0.639….

Why?  What makes hyperelliptic curves over F_3 more often ordinary than their plane curve counterparts?

(Note that the probability of ordinarity, which makes good sense for those who already know Dieudonné modules well, is just the probability that two random maximal isotropic subspaces of a symplectic space over F_q are disjoint.  So some of the computations here are in some sense the “symplectic case” of what Poonen and Rains computed in the orthogonal case.

We compute lots more stuff (distribution of a-numbers, distribution of p-coranks, etc.) and decline to compute a lot more (distribution of Newton polygon, final type…)  Many interesting questions remain!