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!