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

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Elias,  Yara
Max Planck Institute for Mathematics, Max Planck Society;

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arXiv:1708.03652.pdf
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Citation

Celik, T. O., Elias, Y., Gunes, B., Newton, R., Ozman, E., Pries, R., et al. (2018). Non-ordinary curves with a Prym variety of low p-rank. In Women in Numbers Europe II: Contributions to Number Theory and Arithmetic Geometry (pp. 117-158). Cham: Springer.


Cite as: https://hdl.handle.net/21.11116/0000-0003-E249-C
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.