Non-ordinary curves with a Prym variety of low $p$-rank

Abstract

If $\ddot{I}€:Y\to X$ is an unramified double cover of a smooth curve of genus $g$, then the Prym variety $P_{\ddot{I}}€$ is a principally polarized abelian variety of dimension $g-1$. 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 $P_{\ddot{I}}€$ under restrictions on the $p$-rank of $X$. In this paper, when $X$ is a non-hyperelliptic curve of genus $g=3$, we analyze the relationship between the Hasse-Witt matrices of $X$ and $P_{\ddot{I}}€$. As an application, when $p\equiv 5\mathrm{mod}6$, we prove that there exists a curve $X$ of genus $3$ and $p$-rank $f=3$ having an unramified double cover $\ddot{I}€:Y\to X$ for which $P_{\ddot{I}}€$ has $p$-rank $0$ (and is thus supersingular); for $3\le p\le 19$, we verify the same for each $0\le f\le 3$. Using theoretical results about $p$-rank stratifications of moduli spaces, we prove, for small $p$ and arbitrary $g\ge 3$, that there exists an unramified double cover $\ddot{I}€:Y\to X$ such that both $X$ and $P_{\ddot{I}}€$ have small $p$-rank.