## Non-ordinary curves with a Prym variety of low p-rank
Celik, T. O., Elias, Y., Gunes, B., Newton, R. ORCID: https://orcid.org/0000-0003-4925-635X, 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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