The p-rank stratification of Artin-Schreier curves
Annales de l'Institut Fourier, Volume 62 (2012) no. 2, pp. 707-726.

We study a moduli space 𝒜𝒮 g for Artin-Schreier curves of genus g over an algebraically closed field k of characteristic p. We study the stratification of 𝒜𝒮 g by p-rank into strata 𝒜𝒮 g.s of Artin-Schreier curves of genus g with p-rank exactly s. We enumerate the irreducible components of 𝒜𝒮 g,s and find their dimensions. As an application, when p=2, we prove that every irreducible component of the moduli space of hyperelliptic k-curves with genus g and 2-rank s has dimension g-1+s. We also determine all pairs (p,g) for which 𝒜𝒮 g is irreducible. Finally, we study deformations of Artin-Schreier curves with varying p-rank.

Nous étudions un espace de modules 𝒜𝒮 g des courbes d’Artin Schreier de genre g sur k, un corps algébriquement clos de caractéristique p. Nous étudions la stratification de 𝒜𝒮 g par le p-rang, dont la strate 𝒜𝒮 g,s décrit les courbes de genre g et de p-rang s. On énumère les composantes irréductibles de 𝒜𝒮 g,s et on donne leurs dimensions. Une application, dans le cas p=2, est que chaque composante irréductible de l’espace de modules des courbes hyperelliptiques sur k de genre g et de 2-rang s est de dimension g-1+s. Nous déterminons toutes les paires (p,g) pour lesquelles 𝒜𝒮 g est irréductible. Finalement, nous étudions les déformations des courbes d’Artin-Schreier dont le p-rang varie.

DOI: 10.5802/aif.2692
Classification: 11G15, 14H40, 14K15
Keywords: Artin-Schreier, hyperelliptic, curve, moduli, $p$-rank
Mot clés : Artin-Schreier, hyperelliptic, courbe, moduli, $p$-rank

Pries, Rachel 1; Zhu, Hui June 2

1 Colorado State University Mathematics department, Weber 101 Fort Collins, CO, 80523 (USA)
2 SUNY at Buffalo Mathematics department Buffalo, NY, 14260 (USA)
@article{AIF_2012__62_2_707_0,
     author = {Pries, Rachel and Zhu, Hui June},
     title = {The $p$-rank stratification of {Artin-Schreier} curves},
     journal = {Annales de l'Institut Fourier},
     pages = {707--726},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {62},
     number = {2},
     year = {2012},
     doi = {10.5802/aif.2692},
     mrnumber = {2985514},
     zbl = {1281.11062},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2692/}
}
TY  - JOUR
AU  - Pries, Rachel
AU  - Zhu, Hui June
TI  - The $p$-rank stratification of Artin-Schreier curves
JO  - Annales de l'Institut Fourier
PY  - 2012
SP  - 707
EP  - 726
VL  - 62
IS  - 2
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.2692/
DO  - 10.5802/aif.2692
LA  - en
ID  - AIF_2012__62_2_707_0
ER  - 
%0 Journal Article
%A Pries, Rachel
%A Zhu, Hui June
%T The $p$-rank stratification of Artin-Schreier curves
%J Annales de l'Institut Fourier
%D 2012
%P 707-726
%V 62
%N 2
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.2692/
%R 10.5802/aif.2692
%G en
%F AIF_2012__62_2_707_0
Pries, Rachel; Zhu, Hui June. The $p$-rank stratification of Artin-Schreier curves. Annales de l'Institut Fourier, Volume 62 (2012) no. 2, pp. 707-726. doi : 10.5802/aif.2692. https://aif.centre-mersenne.org/articles/10.5802/aif.2692/

[1] Achter, J.; Pries, R. The p -rank strata of the moduli space of hyperelliptic curves (to appear in Advances of Mathematics, arXiv:0902.4637)

[2] Achter, Jeffrey D.; Glass, Darren; Pries, Rachel Curves of given p-rank with trivial automorphism group, Michigan Math. J., Volume 56 (2008) no. 3, pp. 583-592 (arXiv:0708.2199) | DOI | MR

[3] Bertin, J.; Mézard, A. Déformations formelles des revêtements sauvagement ramifiés de courbes algébriques, Invent. Math., Volume 141 (2000) no. 1, pp. 195-238 | DOI | MR

[4] Blache, R. First vertices for generic Newton polygons, and p -cyclic coverings of the projective line (arXiv:0912.2051)

[5] Blache, R. p -Density, exponential sums and Artin-Schreier curves (arXiv:0812.3382)

[6] Crew, R. Étale p-covers in characteristic p, Compositio Math., Volume 52 (1984) no. 1, pp. 31-45 | Numdam | MR | Zbl

[7] Deligne, P.; Mumford, D. The irreducibility of the space of curves of given genus, Inst. Hautes Études Sci. Publ. Math. No., Volume 36 (1969), pp. 75-109 | DOI | Numdam | MR | Zbl

[8] Faber, C.; van der Geer, G. Complete subvarieties of moduli spaces and the Prym map, J. Reine Angew. Math., Volume 573 (2004), pp. 117-137 | DOI | MR

[9] Glass, D.; Pries, R. Hyperelliptic curves with prescribed p-torsion, Manuscripta Math., Volume 117 (2005) no. 3, pp. 299-317 | DOI | MR

[10] Green, Barry; Matignon, Michel Order p automorphisms of the open disc of a p-adic field, J. Amer. Math. Soc., Volume 12 (1999) no. 1, pp. 269-303 | DOI | MR | Zbl

[11] Harbater, D. Moduli of p-covers of curves, Comm. Algebra, Volume 8 (1980) no. 12, pp. 1095-1122 | DOI | MR | Zbl

[12] Hartshorne, Robin Algebraic geometry, Springer-Verlag, New York, 1977 (Graduate Texts in Mathematics, No. 52) | MR | Zbl

[13] Lønsted, K. The hyperelliptic locus with special reference to characteristic two, Math. Ann., Volume 222 (1976) no. 1, pp. 55-61 | DOI | MR | Zbl

[14] Maugeais, S. On a compactification of a Hurwitz space in the wild case (math.AG/0509118)

[15] Maugeais, S. Quelques résultats sur les déformations équivariantes des courbes stables, Manuscripta Math., Volume 120 (2006) no. 1, pp. 53-82 | DOI | MR

[16] Mézard, A. Quelques problèmes de déformations en caractéristique mixte (thèse de doctorat de mathématiques de l’université Joseph Fourier)

[17] Oort, F. Subvarieties of moduli spaces, Invent. Math., Volume 24 (1974), pp. 95-119 | DOI | MR | Zbl

[18] Pries, R. Families of wildly ramified covers of curves, Amer. J. Math., Volume 124 (2002) no. 4, pp. 737-768 | DOI | MR

[19] Scholten, J.; Zhu, H. J. Hyperelliptic curves in characteristic 2, Int. Math. Res. Not. (2002) no. 17, pp. 905-917 | DOI | MR

[20] Sekiguchi, T.; Oort, F.; Suwa, N. On the deformation of Artin-Schreier to Kummer, Ann. Sci. École Norm. Sup. (4), Volume 22 (1989) no. 3, pp. 345-375 | Numdam | MR | Zbl

[21] Serre, J.-P. Corps Locaux, Hermann, 1968 | MR

[22] Völklein, Helmut Groups as Galois groups, Cambridge Studies in Advanced Mathematics, 53, Cambridge University Press, Cambridge, 1996 (An introduction) | MR | Zbl

[23] Zhu, H. J. L-functions of exponential sums over one-dimensional affinoids: Newton over Hodge, Int. Math. Res. Not. (2004) no. 30, pp. 1529-1550 | DOI | MR

[24] Zhu, H. J. Hyperelliptic curves over 𝔽 2 of every 2-rank without extra automorphisms, Proc. Amer. Math. Soc., Volume 134 (2006) no. 2, p. 323-331 (electronic) | DOI | MR

Cited by Sources: