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

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 $g-1$. 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 $P_{\pi }$ 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 $P_{\pi }$. As an application, when $p \equiv 5 \bmod 6$,weprovethatthereexistsacurve$X$ofgenus$3$and$p$-rank$f=3$havinganunramifieddoublecover$\pi:Y \to X$forwhich$P_\pi$has$p$-rank$0$(andisthussupersingular);for$3 \leq p \leq 19$, 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 $g \geq 3$,thatthereexistsanunramifieddoublecover$\pi: Y \to X$suchthatboth$X$and$P_\pi$havesmall$p$-rank.