Non-ordinary Curves with a Prym Variety of Low p-Rank

Abstract

If π: Y → X is an unramified double cover of a smooth curve of genus g, then the Prym variety Pπ 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 Pπ. 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 Pπ 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 Pπ have small p-rank.