The p -rank stratification of Artin-Schreier curves

Rachel Pries[1]; Hui June Zhu[2]

  • [1] Colorado State University Mathematics department, Weber 101 Fort Collins, CO, 80523 (USA)
  • [2] SUNY at Buffalo Mathematics department Buffalo, NY, 14260 (USA)

Annales de l’institut Fourier (2012)

  • Volume: 62, Issue: 2, page 707-726
  • ISSN: 0373-0956

Abstract

top
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.

How to cite

top

Pries, Rachel, and Zhu, Hui June. "The $p$-rank stratification of Artin-Schreier curves." Annales de l’institut Fourier 62.2 (2012): 707-726. <http://eudml.org/doc/251106>.

@article{Pries2012,
abstract = {We study a moduli space $\{\mathcal\{AS\}\}_g$ for Artin-Schreier curves of genus $g$ over an algebraically closed field $k$ of characteristic $p$. We study the stratification of $\{\mathcal\{AS\}\}_g$ by $p$-rank into strata $\{\mathcal\{AS\}\}_\{g.s\}$ of Artin-Schreier curves of genus $g$ with $p$-rank exactly $s$. We enumerate the irreducible components of $\{\mathcal\{AS\}\}_\{g,s\}$ 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 $\{\mathcal\{AS\}\}_g$ is irreducible. Finally, we study deformations of Artin-Schreier curves with varying $p$-rank.},
affiliation = {Colorado State University Mathematics department, Weber 101 Fort Collins, CO, 80523 (USA); SUNY at Buffalo Mathematics department Buffalo, NY, 14260 (USA)},
author = {Pries, Rachel, Zhu, Hui June},
journal = {Annales de l’institut Fourier},
keywords = {Artin-Schreier; hyperelliptic; curve; moduli; $p$-rank; Artin-Schreier curve; moduli space of hyperelliptic curves; p-rank},
language = {eng},
number = {2},
pages = {707-726},
publisher = {Association des Annales de l’institut Fourier},
title = {The $p$-rank stratification of Artin-Schreier curves},
url = {http://eudml.org/doc/251106},
volume = {62},
year = {2012},
}

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
PB - Association des Annales de l’institut Fourier
VL - 62
IS - 2
SP - 707
EP - 726
AB - We study a moduli space ${\mathcal{AS}}_g$ for Artin-Schreier curves of genus $g$ over an algebraically closed field $k$ of characteristic $p$. We study the stratification of ${\mathcal{AS}}_g$ by $p$-rank into strata ${\mathcal{AS}}_{g.s}$ of Artin-Schreier curves of genus $g$ with $p$-rank exactly $s$. We enumerate the irreducible components of ${\mathcal{AS}}_{g,s}$ 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 ${\mathcal{AS}}_g$ is irreducible. Finally, we study deformations of Artin-Schreier curves with varying $p$-rank.
LA - eng
KW - Artin-Schreier; hyperelliptic; curve; moduli; $p$-rank; Artin-Schreier curve; moduli space of hyperelliptic curves; p-rank
UR - http://eudml.org/doc/251106
ER -

References

top
  1. J. Achter, R. Pries, The p -rank strata of the moduli space of hyperelliptic curves Zbl1219.14033
  2. Jeffrey D. Achter, Darren Glass, Rachel Pries, Curves of given p -rank with trivial automorphism group, Michigan Math. J. 56 (2008), 583-592 Zbl1183.14042MR2490647
  3. J. Bertin, A. Mézard, Déformations formelles des revêtements sauvagement ramifiés de courbes algébriques, Invent. Math. 141 (2000), 195-238 Zbl0993.14014MR1767273
  4. R. Blache, First vertices for generic Newton polygons, and p -cyclic coverings of the projective line 
  5. R. Blache, p -Density, exponential sums and Artin-Schreier curves 
  6. R. Crew, Étale p -covers in characteristic p , Compositio Math. 52 (1984), 31-45 Zbl0558.14009MR742696
  7. P. Deligne, D. Mumford, The irreducibility of the space of curves of given genus, Inst. Hautes Études Sci. Publ. Math. No. 36 (1969), 75-109 Zbl0181.48803MR262240
  8. C. Faber, G. van der Geer, Complete subvarieties of moduli spaces and the Prym map, J. Reine Angew. Math. 573 (2004), 117-137 Zbl1075.14023MR2084584
  9. D. Glass, R. Pries, Hyperelliptic curves with prescribed p -torsion, Manuscripta Math. 117 (2005), 299-317 Zbl1093.14039MR2154252
  10. Barry Green, Michel Matignon, Order p automorphisms of the open disc of a p -adic field, J. Amer. Math. Soc. 12 (1999), 269-303 Zbl0923.14007MR1630112
  11. D. Harbater, Moduli of p -covers of curves, Comm. Algebra 8 (1980), 1095-1122 Zbl0471.14011MR579791
  12. Robin Hartshorne, Algebraic geometry, (1977), Springer-Verlag, New York Zbl0531.14001MR463157
  13. K. Lønsted, The hyperelliptic locus with special reference to characteristic two, Math. Ann. 222 (1976), 55-61 Zbl0314.14009MR407030
  14. S. Maugeais, On a compactification of a Hurwitz space in the wild case Zbl1099.14501
  15. S. Maugeais, Quelques résultats sur les déformations équivariantes des courbes stables, Manuscripta Math. 120 (2006), 53-82 Zbl1101.14038MR2223481
  16. A. Mézard, Quelques problèmes de déformations en caractéristique mixte 
  17. F. Oort, Subvarieties of moduli spaces, Invent. Math. 24 (1974), 95-119 Zbl0259.14011MR424813
  18. R. Pries, Families of wildly ramified covers of curves, Amer. J. Math. 124 (2002), 737-768 Zbl1059.14033MR1914457
  19. J. Scholten, H. J. Zhu, Hyperelliptic curves in characteristic 2, Int. Math. Res. Not. (2002), 905-917 Zbl1034.14013MR1899907
  20. T. Sekiguchi, F. Oort, N. Suwa, On the deformation of Artin-Schreier to Kummer, Ann. Sci. École Norm. Sup. (4) 22 (1989), 345-375 Zbl0714.14024MR1011987
  21. J.-P. Serre, Corps Locaux, (1968), Hermann Zbl0137.02501MR354618
  22. Helmut Völklein, Groups as Galois groups, 53 (1996), Cambridge University Press, Cambridge Zbl0868.12003MR1405612
  23. H. J. Zhu, L -functions of exponential sums over one-dimensional affinoids: Newton over Hodge, Int. Math. Res. Not. (2004), 1529-1550 Zbl1089.11044MR2049830
  24. H. J. Zhu, Hyperelliptic curves over 𝔽 2 of every 2-rank without extra automorphisms, Proc. Amer. Math. Soc. 134 (2006), 323-331 (electronic) Zbl1111.14022MR2175998

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.