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

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.
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https://hal.archives-ouvertes.fr/hal-01579544
Contributor : Marie-Annick Guillemer <>
Submitted on : Thursday, August 31, 2017 - 12:07:39 PM
Last modification on : Thursday, November 15, 2018 - 11:56:47 AM

### Identifiers

• HAL Id : hal-01579544, version 1
• ARXIV : 1708.03652

### Citation

Turku Ozlum Celik, Yara Elias, Burcin Gunes, Rachel Newton, Ekin Ozman, et al.. Non-ordinary curves with a Prym variety of low $p$-rank. 2017. ⟨hal-01579544⟩

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