TY - JOUR

T1 - Random Dieudonné modules, random p-divisible groups, and random curves over finite fields

AU - Cais, Bryden

AU - Ellenberg, Jordan S.

AU - Zureick-Brown, David

N1 - Funding Information:
The first and third authors are supported by NSA Young Investigator grants. The second author is supported by an NSF grant and a Romnes Family Fellowship.

PY - 2013/7

Y1 - 2013/7

N2 - 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.

AB - 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.

KW - Algebraic curve

KW - Arithmetic statistics

KW - Dieduonne module

KW - Random matrices

KW - p-divisible group

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U2 - 10.1017/S1474748012000862

DO - 10.1017/S1474748012000862

M3 - Article

AN - SCOPUS:84882262908

SN - 1474-7480

VL - 12

SP - 651

EP - 676

JO - Journal of the Institute of Mathematics of Jussieu

JF - Journal of the Institute of Mathematics of Jussieu

IS - 3

ER -