Advertisement

Newton Polygons of Cyclic Covers of the Projective Line Branched at Three Points

  • Wanlin LiEmail author
  • Elena Mantovan
  • Rachel Pries
  • Yunqing Tang
Conference paper
Part of the Association for Women in Mathematics Series book series (AWMS, volume 19)

Abstract

We review the Shimura–Taniyama method for computing the Newton polygon of an abelian variety with complex multiplication. We apply this method to cyclic covers of the projective line branched at three points. As an application, we produce multiple new examples of Newton polygons that occur for Jacobians of smooth curves in characteristic p. Under certain congruence conditions on p, these include: the supersingular Newton polygon for each genus g with 4 ≤ g ≤ 11; nine non-supersingular Newton polygons with p-rank 0 with 4 ≤ g ≤ 11; and, for all g ≥ 5, the Newton polygon with p-rank g − 5 having slopes 1∕5 and 4∕5.

Keywords

Curve Cyclic cover Jacobian Abelian variety Moduli space Reduction Supersingular Newton polygon p-rank Dieudonné module p-divisible group Complex multiplication Shimura–Taniyama method 

MSC10

Primary 11G20 11M38 14G10 14H40 14K22; Secondary 11G10 14H10 14H30 14H40 

References

  1. 1.
    Revêtements étales et groupe fondamental (SGA 1), Documents Mathématiques (Paris) [Mathematical Documents (Paris)], 3, Société Mathématique de France, Paris, 2003, Séminaire de géométrie algébrique du Bois Marie 1960–61. [Algebraic Geometry Seminar of Bois Marie 1960–61], Directed by A. Grothendieck, With two papers by M. Raynaud, Updated and annotated reprint of the 1971 original [Lecture Notes in Math., 224, Springer, Berlin; MR0354651 (50 #7129)]. MR 2017446 (2004g:14017)Google Scholar
  2. 2.
    Jeffrey D. Achter and Rachel Pries, Generic Newton polygons for curves of givenp-rank, Algebraic curves and finite fields, Radon Ser. Comput. Appl. Math., vol. 16, De Gruyter, Berlin, 2014, pp. 1–21. MR 3287680Google Scholar
  3. 3.
    P. Deligne and G. D. Mostow, Monodromy of hypergeometric functions and nonlattice integral monodromy, Inst. Hautes Études Sci. Publ. Math. (1986), no. 63, 5–89. MR 849651CrossRefGoogle Scholar
  4. 4.
    Carel Faber and Gerard van der Geer, Complete subvarieties of moduli spaces and the Prym map, J. Reine Angew. Math. 573 (2004), 117–137. MR 2084584Google Scholar
  5. 5.
    Josep González, Hasse-Witt matrices for the Fermat curves of prime degree, Tohoku Math. J. (2) 49 (1997), no. 2, 149–163. MR 1447179MathSciNetCrossRefGoogle Scholar
  6. 6.
    Benedict H. Gross and David E. Rohrlich, Some results on the Mordell-Weil group of the Jacobian of the Fermat curve, Invent. Math. 44 (1978), no. 3, 201–224. MR 0491708Google Scholar
  7. 7.
    Taira Honda, On the Jacobian variety of the algebraic curvey 2 = 1 − x lover a field of characteristicp > 0, Osaka J. Math. 3 (1966), 189–194. MR 0225777Google Scholar
  8. 8.
    Ju. I. Manin, Theory of commutative formal groups over fields of finite characteristic, Uspehi Mat. Nauk 18 (1963), no. 6 (114), 3–90. MR 0157972 (28 #1200)Google Scholar
  9. 9.
    James Milne, Complex multiplication, course notes, available at www.jmilne.org/math/CourseNotes.Google Scholar
  10. 10.
    Ben Moonen, Special subvarieties arising from families of cyclic covers of the projective line, Doc. Math. 15 (2010), 793–819. MR 2735989Google Scholar
  11. 11.
    Frans Oort, Abelian varieties isogenous to a Jacobian in problems from the Workshop on Automorphisms of Curves, Rend. Sem. Mat. Univ. Padova 113 (2005), 129–177. MR 2168985 (2006d:14027)Google Scholar
  12. 12.
    Rachel Pries, Current results on Newton polygons of curves, to appear in Questions in Arithmetic Algebraic Geometry, Advanced Lectures in Mathematics, Chapter 6.Google Scholar
  13. 13.
    Rachel Pries, Thep-torsion of curves with largep-rank, Int. J. Number Theory 5 (2009), no. 6, 1103–1116. MR 2569747Google Scholar
  14. 14.
    John Tate, Classes d’isogénie des variétés abéliennes sur un corps fini (d’après T. Honda), Séminaire Bourbaki. Vol. 1968/69: Exposés 347–363, Lecture Notes in Math., vol. 175, Springer, Berlin, 1971, pp. Exp. No. 352, 95–110. MR 3077121Google Scholar
  15. 15.
    Gerard van der Geer and Marcel van der Vlugt, On the existence of supersingular curves of given genus, J. Reine Angew. Math. 458 (1995), 53–61. MR 1310953 (95k:11084)Google Scholar
  16. 16.
    André Weil, Numbers of solutions of equations in finite fields, Bull. Amer. Math. Soc. 55 (1949), 497–508. MR 0029393MathSciNetCrossRefGoogle Scholar
  17. 17.
    Stefan Wewers, Construction of Hurwitz spaces, Dissertation, 1998.Google Scholar
  18. 18.
    Noriko Yui, On the Jacobian variety of the Fermat curve, J. Algebra 65 (1980), no. 1, 1–35. MR 578793MathSciNetCrossRefGoogle Scholar

Copyright information

© The Author(s) and The Association for Women in Mathematics 2019

Authors and Affiliations

  • Wanlin Li
    • 1
    Email author
  • Elena Mantovan
    • 2
  • Rachel Pries
    • 3
  • Yunqing Tang
    • 4
  1. 1.Department of MathematicsUniversity of WisconsinMadisonUSA
  2. 2.Department of MathematicsCalifornia Institute of TechnologyPasadenaUSA
  3. 3.Department of MathematicsColorado State UniversityFort CollinsUSA
  4. 4.Department of MathematicsPrinceton UniversityPrincetonUSA

Personalised recommendations