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

Abstract

Link to this article: http://journals.cambridge.org/abstract_S1474748012000862How to cite this article: Bryden Cais, Jordan S. Ellenberg and David Zureick-Brown (2013). Random Dieudonné modules, random -divisible groups, and random curves over nite elds.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 F 3 appear substantially less likely to be ordinary than hyperelliptic curves over F 3 .

The Distribution of Torsion Subschemes of Abelian varieties

“…the error is bounded by D/ √ q for some constant D which depends only on g, n and p. Thanks to Lemma 2.2, this is compatible with the broad philosophy of [5], and even that of [9]; a (polarized) group (scheme) occurs with frequency inversely proportional to the size of its automorphism group.…”

“…The present investigation was inspired by the work of Cais, Ellenberg and Zureick-Brown, even though the results here are incomparable with those of [5]. The author acknowledges helpful discussions with Oort and Liedtke, especially concerning the example in Section 5.3.…”

Computing quadratic function fields with high 3-rank via cubic field tabulation

“…Some recent work has been done on generating data and developing a heuristic for the case where the prime p for which the p-rank is under consideration is equal to the characteristic of the field. See [7] for some new results along these lines.…”

Curves in characteristic $2$ with non-trivial $2$-torsion

“…if and only if D ∼ Θ geom . But the proof of Theorem 2 (or of [CEZB,Thm. 4.2]) is precisely about showing that if c i,j = 0 for some i and j that are both odd, then Θ arith ∼ Θ geom .…”

“…The starting point of this article is a recent theorem by Cais, Ellenberg and Zureick-Brown [CEZB,Thm. 4.2], asserting that over a finite field k of characteristic 2, almost all smooth plane projective curves of a given odd degree d ≥ 3 have a non-trivial krational 2-torsion point on their Jacobian.…”