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

@article{Celik2017NonordinaryCW,
  title={Non-ordinary curves with a Prym variety of low \$p\$-rank},
  author={Turku Ozlum Celik and Yara Elias and Burcin Gunes and Rachel Newton and Ekin Ozman and Rachel J. Pries and Lara Thomas},
  journal={arXiv: Number Theory},
  year={2017}
}
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… Expand
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References

SHOWING 1-10 OF 24 REFERENCES
Ordinary and almost ordinary Prym varieties
We study the relationship between the $p$-rank of a curve and the $p$-ranks of the Prym varieties of its cyclic covers in characteristic $p >0$. For arbitrary $p$, $g \geq 3$ and $0 \leq f \leq g$,Expand
Superspecial curves of genus 4 in small characteristic
TLDR
It is proved that there does not exist a superspecial curve of genus $4$ in characteristic $7$, which implies the non-existence of maximal curves of genus$4$ over $\mathbb{F}_{49}$, which updates the table at {\tt this http URL}. Expand
Syzygies of torsion bundles and the geometry of the level ℓ modular variety over $\overline{\mathcal{M}}_{g}$
We formulate, and in some cases prove, three statements concerning the purity or, more generally, the naturality of the resolution of various modules one can attach to a generic curve of genus g andExpand
The arithmetic of Prym varieties in genus 3
  • N. Bruin
  • Mathematics
  • Compositio Mathematica
  • 2008
Abstract Given a curve of genus 3 with an unramified double cover, we give an explicit description of the associated Prym variety. We also describe how an unramified double cover of aExpand
On the Jacobian varieties of hyperelliptic curves over fields of characteristic p > 2
It is well known that an Abelian variety X of dimension g defined over a field K of characteristic p > 0 yields a p-divisible group X(p) of dimension g and of height 2g. Let I’ be the formal groupExpand
The p-Rank of Ramified Covers of Curves
  • I. Bouw
  • Mathematics
  • Compositio Mathematica
  • 2001
In this paper we study the p-rank of Abelian prime-to-p covers of the generic r-pointed curve of genus g. There is an obvious bound on the p-rank of the cover. We show that it suffices to compute theExpand
Syzygies of torsion bundles and the geometry of the level l modular variety over Mg
We formulate, and in some cases prove, three statements concerning the purity or, more generally the naturality of the resolution of various rings one can attach to a generic curve of genus g and aExpand
Complete subvarieties of moduli spaces and the Prym map
We prove that in characteristic p>0 the locus of stable curves of p-rank at most f is pure of codimension g-f in the moduli space of stable curves. Then we consider the Prym map and analyze it usingExpand
Subvarieties of moduli spaces
In this paper we try to decide along algebraic lines whether moduli spaces of abelian varieties or of algebraic curves contain complete subvarieties. In Theorem (1.1) we consider abelian varietiesExpand
MONODROMY OF THE p-RANK STRATA OF THE MODULI SPACE OF CURVES
We determine the -monodromy and -monodromy of every irreducible component of the stratum of curves of genus g and p-rank f in characteristic p. In particular, we prove that the -monodromy of everyExpand
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