Random Dieudonné modules, random $p$-divisible groups, and random curves over finite fields. (July 2013) - Cais, Bryden; Ellenberg, Jordan S.; Zureick-Brown, David
Abstract We describe a probability distribution on isomorphism classes of principally quasi-polarized <alternatives><inline-graphic xlink:href="ark:/27927/pgh350fxmrj" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$p$]]> </alternatives>-divisible groups over a finite field <alternatives><inline-graphic xlink:href="ark:/27927/pgh350fxmks" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$k$]]> </alternatives> of characteristic <alternatives><inline-graphic xlink:href="ark:/27927/pgh350fxksk" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$p$]]> </alternatives> which can reasonably be thought of as a 'uniform distribution', and we compute the distribution of various statistics (<alternatives><inline-graphic xlink:href="ark:/27927/pgh350fxmhp" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$p$]]> </alternatives>-corank, <alternatives><inline-graphic xlink:href="ark:/27927/pgh350fxm1z" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$a$]]> </alternatives>-number, etc.) of <alternatives><inline-graphic xlink:href="ark:/27927/pgh350fxmtn" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$p$]]> </alternatives>-divisible groups drawn from this distribution. It is then natural to ask to what extent the <alternatives><inline-graphic xlink:href="ark:/27927/pgh350fxkt4" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$p$]]> </alternatives>-divisible groups attached to a randomly chosen hyperelliptic curve (respectively, curve; respectively, abelian variety) over <alternatives><inline-graphic xlink:href="ark:/27927/pgh350fxmpf" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$k$]]> </alternatives> are uniformly distributed in this sense. This heuristic is analogous to conjectures of Cohen–Lenstra type for <alternatives><inline-graphic xlink:href="ark:/27927/pgh350fxms3" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$\text{char~} k\not = p$]]> </alternatives>, in which case the random <alternatives><inline-graphic xlink:href="ark:/27927/pgh350fxmx9" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$p$]]> </alternatives>-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 <alternatives><inline-graphic xlink:href="ark:/27927/pgh350fxmnw" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[${\mathbf{F} }_{3} $]]> </alternatives> appear substantially less likely to be ordinary than hyperelliptic curves over <alternatives><inline-graphic xlink:href="ark:/27927/pgh350fxm0d" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[${\mathbf{F} }_{3} $]]> </alternatives>.