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The prank stratification of ArtinSchreier curves
"... ABSTRACT. We study a moduli spaceASg for ArtinSchreier curves of genus g over an algebraically closed field k of characteristic p. We study the stratification ofASg by prank into strataASg.s of ArtinSchreier curves of genus g with prank exactly s. We enumerate the irreducible components ofASg,s ..."
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ABSTRACT. We study a moduli spaceASg for ArtinSchreier curves of genus g over an algebraically closed field k of characteristic p. We study the stratification ofASg by prank into strataASg.s of ArtinSchreier curves of genus g with prank exactly s. We enumerate the irreducible components ofASg,s and find their dimensions. As an application, when p = 2, we prove that every irreducible component of the moduli space of hyperelliptic kcurves with genus g and 2rank s has dimension g−1+s. We also determine all pairs (p,g) for whichASg is irreducible. Finally, we study deformations of ArtinSchreier curves with varying prank. La stratification de prang des courbes d’ArtinSchreier RÉSUMÉ. Nous étudions un espace de modulesASg des courbes d’Artin Schreier de genre g sur k, un corps algébriquement clos de caractéristique p. Nous étudions la stratification deASg par le prang, dont la strateASg,s décrit les courbes de genre g et de prang s. On énumère les composantes irréductibles deASg,s et on donne leurs dimensions. Une application, dans le cas p = 2, est que chaque composante irréductible de l’espace de modules des courbes hyperelliptiques sur k de genre g et de 2rang s est de dimension g−1+s. Finalement, nous déterminons toutes les paires (p,g) pour lesquellesASg est irréductible. 1.
Random Dieudonné modules, random pdivisible groups, and random curves over finite fields
, 2012
"... We describe a probability distribution on isomorphism classes of principally quasipolarized pdivisible groups over a finite field k of characteristic p which can reasonably be thought of as “uniform distribution, ” and we compute the distribution of various statistics (pcorank, anumber, etc.) of ..."
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We describe a probability distribution on isomorphism classes of principally quasipolarized pdivisible groups over a finite field k of characteristic p which can reasonably be thought of as “uniform distribution, ” and we compute the distribution of various statistics (pcorank, anumber, etc.) of pdivisible groups drawn from this distribution. It is then natural to ask to what extent the pdivisible groups attached to a randomly chosen hyperelliptic curve (resp. curve, resp. abelian variety) over k are uniformly distributed in this sense. This heuristic is analogous to conjectures of CohenLenstra type for char k = p, in which case the random pdivisible 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 F3. 1