Random Dieudonné modules, random $p$-divisible groups, and random curves over finite fields. (July 2013)

• Record Type: Journal Article
• Authors: Cais, Bryden; Ellenberg, Jordan S.; Zureick-Brown, David
• Is Part Of: Journal of the Institute of Mathematics of Jussieu. Volume 12:Number 3(2013)
• Page Start: 651
• Page End: 676
• Publication Date: 2013-07
• Subjects: Mathematics -- Periodicals; 510.5
• DOI: 10.1017/S1474748012000862
• Languages: English
• ISSNs: 1474-7480

Abstract

<abstract abstract-type="normal"> <title>Abstract</title> <p>We describe a probability distribution on isomorphism classes of principally quasi-polarized <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgh350fxmrj" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$p$]]></tex-math></alternatives></inline-formula>-divisible groups over a finite field <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgh350fxmks" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$k$]]></tex-math></alternatives></inline-formula> of characteristic <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgh350fxksk" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$p$]]></tex-math></alternatives></inline-formula> which can reasonably be thought of as a 'uniform distribution', and we compute the distribution of various statistics (<inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgh350fxmhp" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$p$]]></tex-math></alternatives></inline-formula>-corank, <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgh350fxm1z" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$a$]]></tex-math></alternatives></inline-formula>-number, etc.) of <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgh350fxmtn" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$p$]]></tex-math></alternatives></inline-formula>-divisible groups drawn from this distribution. It is then natural to ask to what extent the <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgh350fxkt4" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$p$]]></tex-math></alternatives></inline-formula>-divisible groups attached to a randomly chosen hyperelliptic curve (respectively, curve; respectively, abelian variety) over <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgh350fxmpf" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$k$]]></tex-math></alternatives></inline-formula> are uniformly distributed in this sense. This heuristic is analogous to conjectures of Cohen–Lenstra type for <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgh350fxms3" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$\text{char~} k\not = p$]]></tex-math></alternatives></inline-formula>, in which case the random <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgh350fxmx9" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$p$]]></tex-math></alternatives></inline-formula>-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 <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgh350fxmnw" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[${\mathbf{F} }_{3} $]]></tex-math></alternatives></inline-formula> appear substantially less likely to be ordinary than hyperelliptic curves over <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgh350fxm0d" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[${\mathbf{F} }_{3} $]]></tex-math></alternatives></inline-formula>.</p> </abstract>