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

TL;DR

This paper investigates the possible $p$-ranks of Prym varieties associated with unramified double covers of genus 3 curves over fields of characteristic $p$, revealing new examples of low $p$-rank Prym varieties and their relation to the base curve.

Contribution

It provides explicit constructions and theoretical results showing the existence of unramified double covers with Prym varieties of low $p$-rank, especially in characteristic $p ot o 0$, extending understanding of $p$-rank stratifications.

Findings

Existence of genus 3 curves with $p$-rank 3 and Prym of $p$-rank 0 for certain primes.

Verification of similar phenomena for all $0 \\leq f \\leq 3$ when $3 \\leq p \\leq 19$.

General results on small $p$-rank Prym varieties for higher genus and small primes.

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 $g-1$. 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 $p \equiv 5 \bmod 6$, 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.