Random Dieudonné modules, random $p$-divisible groups, and random curves over finite fields. (July 2013) - Cais, Bryden
Abstract We describe a probability distribution on isomorphism classes of principally quasi-polarized $p$ -divisible groups over a finite field $k$ of characteristic $p$ which can reasonably be thought of as a 'uniform distribution', and we compute the distribution of various statistics ( $p$ -corank, $a$ -number, etc.) of $p$ -divisible groups drawn from this distribution. It is then natural to ask to what extent the $p$ -divisible groups attached to a randomly chosen hyperelliptic curve (respectively, curve; respectively, abelian variety) over $k$ are uniformly distributed in this sense. This heuristic is analogous to conjectures of Cohen–Lenstra type for $\text{char~} k\not = p$ , in which case the random $p$ -divisible group is defined by a random matrix recording the action of Frobenius. Extensive numerical investigation reveals some cases of agreement with the heuristic and some interesting discrepancies. For example, plane curves over ${\mathbf{F} }_{3} $ appear substantially less likely to be ordinary than hyperelliptic curves over ${\mathbf{F} }_{3} $ .