Abstract/Summary

If π:Y→X is an unramified double cover of a smooth curve of genus g, then the Prym variety P_π 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 Pπ 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_π. As an application, when p≡5 mod 6, we prove that there exists a curve X of genus 3 and p-rank f=3 having an unramified double cover π:Y→X for which P_π has p-rank 0 (and is thus supersingular); for 3≤p≤19, we verify the same for each 0≤f≤3. Using theoretical results about p-rank stratifications of moduli spaces, we prove, for small p and arbitrary g≥3, that there exists an unramified double cover π:Y→X such that both X and P_π have small p-rank.