Nonordinary curves with a Prym variety of low $p$rank
Abstract
If $\pi: Y \to X$ is an unramified double cover of a smooth curve of genus $g$, then the Prym variety $P_\pi$ is a principally polarized abelian variety of dimension $g1$. When $X$ is defined over an algebraically closed field $k$ of characteristic $p$, it is not known in general which $p$ranks can occur for $P_\pi$ under restrictions on the $p$rank of $X$. In this paper, when $X$ is a nonhyperelliptic curve of genus $g=3$, we analyze the relationship between the HasseWitt matrices of $X$ and $P_\pi$. As an application, when $p \equiv 5 \bmod 6$, we prove that there exists a curve $X$ of genus $3$ and $p$rank $f=3$ having an unramified double cover $\pi:Y \to X$ for which $P_\pi$ has $p$rank $0$ (and is thus supersingular); for $3 \leq p \leq 19$, we verify the same for each $0 \leq f \leq 3$. Using theoretical results about $p$rank stratifications of moduli spaces, we prove, for small $p$ and arbitrary $g \geq 3$, that there exists an unramified double cover $\pi: Y \to X$ such that both $X$ and $P_\pi$ have small $p$rank.
 Publication:

arXiv eprints
 Pub Date:
 August 2017
 arXiv:
 arXiv:1708.03652
 Bibcode:
 2017arXiv170803652O
 Keywords:

 Mathematics  Number Theory;
 Mathematics  Algebraic Geometry;
 11G20;
 14H10;
 14H40;
 14K15;
 14Q05