We study a moduli space for Artin-Schreier curves of genus over an algebraically closed field of characteristic . We study the stratification of by -rank into strata of Artin-Schreier curves of genus with -rank exactly . We enumerate the irreducible components of and find their dimensions. As an application, when , we prove that every irreducible component of the moduli space of hyperelliptic -curves with genus and -rank has dimension . We also determine all pairs for which is irreducible. Finally, we study deformations of Artin-Schreier curves with varying -rank.
Nous étudions un espace de modules des courbes d’Artin Schreier de genre sur , un corps algébriquement clos de caractéristique . Nous étudions la stratification de par le -rang, dont la strate décrit les courbes de genre et de -rang . On énumère les composantes irréductibles de et on donne leurs dimensions. Une application, dans le cas , est que chaque composante irréductible de l’espace de modules des courbes hyperelliptiques sur de genre et de -rang est de dimension . Nous déterminons toutes les paires pour lesquelles est irréductible. Finalement, nous étudions les déformations des courbes d’Artin-Schreier dont le -rang varie.
Keywords: Artin-Schreier, hyperelliptic, curve, moduli, $p$-rank
Mot clés : Artin-Schreier, hyperelliptic, courbe, moduli, $p$-rank
Pries, Rachel 1; Zhu, Hui June 2
@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] The -rank strata of the moduli space of hyperelliptic curves (to appear in Advances of Mathematics, arXiv:0902.4637)
[2] Curves of given -rank with trivial automorphism group, Michigan Math. J., Volume 56 (2008) no. 3, pp. 583-592 (arXiv:0708.2199) | DOI | MR
[3] Déformations formelles des revêtements sauvagement ramifiés de courbes algébriques, Invent. Math., Volume 141 (2000) no. 1, pp. 195-238 | DOI | MR
[4] First vertices for generic Newton polygons, and -cyclic coverings of the projective line (arXiv:0912.2051)
[5] -Density, exponential sums and Artin-Schreier curves (arXiv:0812.3382)
[6] Étale -covers in characteristic , Compositio Math., Volume 52 (1984) no. 1, pp. 31-45 | Numdam | MR | Zbl
[7] 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] Complete subvarieties of moduli spaces and the Prym map, J. Reine Angew. Math., Volume 573 (2004), pp. 117-137 | DOI | MR
[9] Hyperelliptic curves with prescribed -torsion, Manuscripta Math., Volume 117 (2005) no. 3, pp. 299-317 | DOI | MR
[10] Order automorphisms of the open disc of a -adic field, J. Amer. Math. Soc., Volume 12 (1999) no. 1, pp. 269-303 | DOI | MR | Zbl
[11] Moduli of -covers of curves, Comm. Algebra, Volume 8 (1980) no. 12, pp. 1095-1122 | DOI | MR | Zbl
[12] Algebraic geometry, Springer-Verlag, New York, 1977 (Graduate Texts in Mathematics, No. 52) | MR | Zbl
[13] The hyperelliptic locus with special reference to characteristic two, Math. Ann., Volume 222 (1976) no. 1, pp. 55-61 | DOI | MR | Zbl
[14] On a compactification of a Hurwitz space in the wild case (math.AG/0509118)
[15] Quelques résultats sur les déformations équivariantes des courbes stables, Manuscripta Math., Volume 120 (2006) no. 1, pp. 53-82 | DOI | MR
[16] Quelques problèmes de déformations en caractéristique mixte (thèse de doctorat de mathématiques de l’université Joseph Fourier)
[17] Subvarieties of moduli spaces, Invent. Math., Volume 24 (1974), pp. 95-119 | DOI | MR | Zbl
[18] Families of wildly ramified covers of curves, Amer. J. Math., Volume 124 (2002) no. 4, pp. 737-768 | DOI | MR
[19] Hyperelliptic curves in characteristic 2, Int. Math. Res. Not. (2002) no. 17, pp. 905-917 | DOI | MR
[20] 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] Corps Locaux, Hermann, 1968 | MR
[22] Groups as Galois groups, Cambridge Studies in Advanced Mathematics, 53, Cambridge University Press, Cambridge, 1996 (An introduction) | MR | Zbl
[23] -functions of exponential sums over one-dimensional affinoids: Newton over Hodge, Int. Math. Res. Not. (2004) no. 30, pp. 1529-1550 | DOI | MR
[24] Hyperelliptic curves over 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: