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

TL;DR

This paper introduces a probability distribution on p-divisible groups over finite fields, computes various statistics, and investigates how well these models match the distribution of Jacobians of random algebraic curves, revealing partial agreement and notable discrepancies.

Contribution

It defines a 'uniform' distribution on isomorphism classes of p-divisible groups and explores its applicability to the Jacobians of random algebraic curves over finite fields.

Findings

Distribution of p-divisible groups matches some heuristics

Discrepancies observed in the likelihood of curves being ordinary

Plane curves over F_3 less likely to be ordinary than hyperelliptic ones

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 "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 (resp. curve, resp. abelian variety) over k are uniformly distributed in this sense. For instance, one can ask whether the proportion of genus-g curves over F_p whose Jacobian is ordinary approaches the limit that such a heuristic would predict. This heuristic is analogous to conjectures of Cohen-Lenstra type for fields k of characteristic other than 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 F_3 appear substantially less likely to be ordinary than hyperelliptic curves over F_3.