## Non-ordinary curves with a Prym variety of low p-rank
Celik, T. O., Elias, Y., Gunes, B., Newton, R., Ozman, E., Pries, R. and Thomas, L.
(2018)
It is advisable to refer to the publisher's version if you intend to cite from this work. See Guidance on citing. ## Abstract/SummaryIf 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.
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