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Random Dieudonné modules, random
$p$-divisible groups, and random curves over finite fields
- Part of
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- Journal:
- Journal of the Institute of Mathematics of Jussieu / Volume 12 / Issue 3 / July 2013
- Published online by Cambridge University Press:
- 13 February 2013, pp. 651-676
- Print publication:
- July 2013
-
- Article
- Export citation
-
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} $.
