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

@article{Cais2013RandomDM, title={Random Dieudonn{\'e} modules, random \$p\$-divisible groups, and random curves over finite fields}, author={Bryden R. Cais and Jordan S. Ellenberg and David Zureick-Brown}, journal={Journal of the Institute of Mathematics of Jussieu}, year={2013}, volume={12}, pages={651 - 676} }

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… Expand

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#### 9 Citations

Proportion of ordinarity in some families of curves over finite fields

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A curve over a field of characteristic $p$ is called ordinary if the $p$-torsion of its Jacobian as large as possible, that is, an $\mathbb{F}_p$ vector space of dimension equal to its genus. In this… Expand

THE DISTRIBUTION OF TORSION SUBSCHEMES OF ABELIAN VARIETIES

- Mathematics
- Journal of the Institute of Mathematics of Jussieu
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We consider the distribution of $p$ -power group schemes among the torsion of abelian varieties over finite fields of characteristic $p$ , as follows. Fix natural numbers $g$ and $n$ , and let… Expand

Curves in characteristic 2 with non-trivial 2-torsion

- Computer Science, Mathematics
- Adv. Math. Commun.
- 2014

This paper extends the observation ofais, Ellenberg and Zureick-Brown that over finite fields of characteristic two, all sufficiently general smooth plane projective curves of a given odd degree admit a non-trivial rational $2-torsion point on their Jacobian to curves given by Laurent polynomials with a fixed Newton polygon. Expand

Smooth curves in toric surfaces (Les courbes lisses dans les surfaces toriques)

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(assumed to be two-dimensional) of f . The earliest such connection is surprisingly old, dating back to 1893, when Baker observed [Bak93] that the geometric genus of Uf is bounded by the number of… Expand

Counting Semilinear Endomorphisms Over Finite Fields

- Mathematics
- 2011

For a finite field k and a triple of integers g \ge r \ge s \ge 0, we count the number of semilinear endomorphisms of a g-dimensional k-vector space which have rank r and stable rank s. Such… Expand

Hasse-Witt and Cartier-Manin matrices: A warning and a request

- Mathematics
- 2017

Let X be a curve in positive characteristic. The Hasse-Witt matrix represents the action of the Frobenius operator on the cohomology group H^1(X,O_X). The Cartier-Manin matrix represents the action… Expand

a-Numbers in Artin-Schreier Covers

- Mathematics
- 2018

Let $\pi : Y \to X$ be a branched $\mathbf{Z}/p \mathbf{Z}$-cover of smooth, projective, geometrically connected curves over a perfect field of characteristic $p>0$. We investigate the relationship… Expand

a-Numbers of Curves in Artin-Schreier Covers.

- Mathematics
- 2018

Let $\pi : Y \to X$ be a branched $\mathbf{Z}/p \mathbf{Z}$-cover of smooth, projective, geometrically connected curves over a perfect field of characteristic $p>0$. We investigate the relationship… Expand

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