×

Ekedahl-Oort strata on the moduli space of curves of genus four. (English) Zbl 1469.11209

Summary: We study the induced Ekedahl-Oort stratification on the moduli of curves of genus 4 in positive characteristic by computing the de Rham cohomology of curves. For moduli space of hyperelliptic curves of genus 4 in characteristic 3, we determine the dimension and reducibility of Ekedahl-Oort strata. For moduli space of curves of genus 4 in odd characteristic, we show the existence of certain Ekedahl-Oort strata and discuss the existence of superspecial curves.

MSC:

11G20 Curves over finite and local fields
14F40 de Rham cohomology and algebraic geometry
14H05 Algebraic functions and function fields in algebraic geometry
14H40 Jacobians, Prym varieties
PDF BibTeX XML Cite
Full Text: DOI arXiv Euclid

References:

[1] J. D. Achter and E. W. Howe, “Hasse-Witt and Cartier-Manin matrices: a warning and a request”, pp. 1-18 in Arithmetic geometry: computation and applications, edited by Y. Aubry et al., Contemp. Math. 722, Amer. Math. Soc., Providence, RI, 2019. · Zbl 1439.11145
[2] C.-L. Chai and F. Oort, “Monodromy and irreducibility of leaves”, Ann. of Math. \((2) 173\):3 (2011), 1359-1396. · Zbl 1228.14021
[3] S. Devalapurkar and J. Halliday, “The Dieudonné modules and Ekedahl-Oort types of Jacobians of hyperelliptic curves in odd characteristic”, 2017.
[4] T. Ekedahl and G. van der Geer, “Cycle classes of the E-O stratification on the moduli of abelian varieties”, pp. 567-636 in Algebra, arithmetic, and geometry: in honor of Yu. I. Manin., Vol. I, edited by Y. Tschinkel and Y. Zarhin, Progr. Math. 269, Birkhäuser, Boston, 2009. · Zbl 1200.14089
[5] A. Elkin, “The rank of the Cartier operator on cyclic covers of the projective line”, J. Algebra 327 (2011), 1-12. · Zbl 1216.14026
[6] A. Elkin and R. Pries, “Ekedahl-Oort strata of hyperelliptic curves in characteristic 2”, Algebra Number Theory 7:3 (2013), 507-532. · Zbl 1282.11065
[7] S. Frei, “The \(s\)-number of hyperelliptic curves”, pp. 107-116 in Women in numbers Europe II, edited by I. I. Bouw et al., Assoc. Women Math. Ser. 11, Springer, 2018. · Zbl 1445.11052
[8] G. van der Geer, “Cycles on the moduli space of abelian varieties”, pp. 65-89 in Moduli of curves and abelian varieties, edited by C. Faber and E. Looijenga, Vieweg Verlag, Braunschweig, Germany, 1999. · Zbl 0974.14031
[9] D. Glass and R. Pries, “Hyperelliptic curves with prescribed \(p\)-torsion”, Manuscripta Math. 117:3 (2005), 299-317. · Zbl 1093.14039
[10] B. Köck and J. Tait, “On the de-Rham cohomology of hyperelliptic curves”, Res. Number Theory 4:2 (2018), art. id. 19, 17 pp. · Zbl 1470.14044
[11] M. Kudo, “On the existence of superspecial nonhyperelliptic curves of genus \(4\)”, 2018.
[12] W. Li, E. Mantovan, R. Pries, and Y. Tang, “Newton polygons of cyclic covers of the projective line branched at three points”, 2018. · Zbl 1460.11088
[13] W. Li, E. Mantovan, R. Pries, and Y. Tang, “Newton polygons arising from special families of cyclic covers of the projective line”, Res. Number Theory 5:1 (2019), art. id. 12, 31 pp. · Zbl 1460.11088
[14] B. Moonen, “Special subvarieties arising from families of cyclic covers of the projective line”, Doc. Math. 15 (2010), 793-819. · Zbl 1236.11056
[15] T. Oda, “The first de Rham cohomology group and Dieudonné modules”, Ann. Sci. École Norm. Sup. \((4) 2 (1969), 63-135\). · Zbl 0175.47901
[16] F. Oort, “A stratification of a moduli space of polarized abelian varieties in positive characteristic”, pp. 47-64 in Moduli of curves and abelian varieties, edited by C. Faber and E. Looijenga, Vieweg Verlag, Braunschweig, Germany, 1999. · Zbl 0974.14030
[17] R. Pries, “The \(p\)-torsion of curves with large \(p\)-rank”, Int. J. Number Theory 5:6 (2009), 1103-1116. · Zbl 1195.11081
[18] J.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.