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

Bryden Cais, Jordan S. Ellenberg, David Zureick-Brown

Research output: Contribution to journalArticle

4 Scopus citations

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 char k ≠ 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 F3 appear substantially less likely to be ordinary than hyperelliptic curves over 3.

Original languageEnglish (US)
Pages (from-to)651-676
Number of pages26
JournalJournal of the Institute of Mathematics of Jussieu
Volume12
Issue number3
DOIs
StatePublished - Jul 1 2013

Keywords

  • Algebraic curve
  • Arithmetic statistics
  • Dieduonne module
  • Random matrices
  • p-divisible group

ASJC Scopus subject areas

  • Mathematics(all)

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