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Newton polygons arising from special families of cyclic covers of the projective line

  • Wanlin Li
  • Elena Mantovan
  • Rachel Pries
  • Yunqing TangEmail author
Research

Abstract

By a result of Moonen, there are exactly 20 positive-dimensional families of cyclic covers of the projective line for which the Torelli image is open and dense in the associated Shimura variety. For each of these, we compute the Newton polygons, and the \(\mu \)-ordinary Ekedahl–Oort type, occurring in the characteristic p reduction of the Shimura variety. We prove that all but a few of the Newton polygons appear on the open Torelli locus. As an application, we produce multiple new examples of Newton polygons and Ekedahl–Oort types of Jacobians of smooth curves in characteristic p. Under certain congruence conditions on p, these include: the supersingular Newton polygon for genus 5, 6, 7; fourteen new non-supersingular Newton polygons for genus 5–7; eleven new Ekedahl–Oort types for genus 4–7 and, for all \(g \ge 6\), the Newton polygon with p-rank \(g-6\) with slopes 1 / 6 and 5 / 6.

Keywords

Curve Cyclic cover Jacobian Abelian variety Shimura variety PEL-type Moduli space Reduction p-Rank Supersingular Newton polygon p-Divisible group Kottwitz method Dieudonné module Ekedahl–Oort type 

Mathematics Subject Classification

Primary 11G18 11G20 11M38 14G10 14G35 Secondary 11G10 14H10 14H30 14H40 14K22 

Notes

Acknowlegements

This project began at the Women in Numbers 4 workshop at the Banff International Research Station. Pries was partially supported by NSF grant DMS-15-02227. We thank Liang Xiao, Xinwen Zhu, and Rong Zhou for discussions about the appendix and thank Liang Xiao for the detailed suggestions on the writing of the appendix. We would like to thank the referee for many helpful comments.

References

  1. 1.
    Achter, J.D., Pries, R.: Monodromy of the \(p\)-rank strata of the moduli space of curves. Int. Math. Res. Not. IMRN, no. 15, Art. ID rnn053, 25 (2008)Google Scholar
  2. 2.
    Achter, J.D., Pries, R.: The \(p\)-rank strata of the moduli space of hyperelliptic curves. Adv. Math. 227(5), 1846–1872 (2011)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Achter, J.D., Pries, R.: Generic Newton polygons for curves of given \(p\)-rank. In: Algebraic Curves and Finite Fields, Radon Ser. Comput. Appl. Math., vol. 16, pp. 1–21. De Gruyter, Berlin (2014)Google Scholar
  4. 4.
    Bouw, I.I.: The \(p\)-rank of ramified covers of curves. Compos. Math. 126(3), 295–322 (2001)MathSciNetCrossRefGoogle Scholar
  5. 5.
    de Jong, A.J., Oort, F.: Purity of the stratification by Newton polygons. J. Am. Math. Soc. 13(1), 209–241 (2000)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Deligne, P., Mostow, G.D.: Monodromy of hypergeometric functions and nonlattice integral monodromy. Inst. Hautes Études Sci. Publ. Math. no. 63, pp. 5–89 (1986)Google Scholar
  7. 7.
    Eischen, E., Mantovan, E.: \(p\)-adic families of automorphic forms in the \(\mu \)-ordinary setting. preprint, available on arXiv:1710.01864
  8. 8.
    Ekedahl, T.: Boundary Behaviour of Hurwitz Schemes, The Moduli Space of Curves (Texel Island, 1994), Progr. Math., vol. 129. Birkhäuser Boston, Boston, MA, pp. 173–198 (1995)Google Scholar
  9. 9.
    Faber, C., van der Geer, G.: Complete subvarieties of moduli spaces and the Prym map. J. Reine Angew. Math. 573, 117–137 (2004)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Fulton, W.: Hurwitz schemes and irreducibility of moduli of algebraic curves. Ann. Math. (2) 90, 542–575 (1969)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Gan, W.T., Hanke, J.P., Yu, J.-K.: On an exact mass formula of Shimura. Duke Math. J. 107(1), 103–133 (2001)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Glass, D., Pries, R.: Hyperelliptic curves with prescribed \(p\)-torsion. Manuscripta Math. 117(3), 299–317 (2005)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Hamacher, P.: The almost product structure of Newton strata in the deformation space of a Barsotti-Tate group with crystalline Tate tensors. Math. Z. 287(3–4), 1255–1277 (2017)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Harris, M., Taylor, R.: The Geometry and Cohomology of Some Simple Shimura Varieties. Annals of Mathematics Studies, vol. 151. Princeton University Press, Princeton, NJ (2001) (With an appendix by Vladimir G. Berkovich)Google Scholar
  15. 15.
    He, X., Rapoport, M.: Stratifications in the reduction of Shimura varieties. Manuscripta Math. 152(3–4), 317–343 (2017)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Kottwitz, R., Rapoport, M.: On the existence of \(F\)-crystals. Comment. Math. Helv. 78(1), 153–184 (2003)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Kottwitz, R.E.: Isocrystals with additional structure. Compos. Math. 56(2), 201–220 (1985)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Kottwitz, R.E.: Tamagawa numbers. Ann. Math. (2) 127(3), 629–646 (1988)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Kottwitz, R.E.: Isocrystals with additional structure. II. Compos. Math. 109(3), 255–339 (1997)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Lau, E.: Smoothness of the truncated display functor. J. Am. Math. Soc. 26(1), 129–165 (2013)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Li, W., Mantovan, E., Pries, R., Tang, Y.: Newton polygon stratification of the Torelli locus in PEL-type Shimura varieties. arXiv:1811.00604
  22. 22.
    Li, W., Mantovan, E., Pries, R., Tang, Y.: Newton polygons of cyclic covers of the projective line branched at three points. Research Directions in Number Theory: Women in Numbers IV (to appear) arXiv:1805.04598
  23. 23.
    Liu, Y., Tian, Y., Xiao, L., Zhang, W., Zhu, X.: On the Beilinson–Bloch–Kato conjecture for Rankin–Selberg (in preparation)Google Scholar
  24. 24.
    Moonen, B.: Serre-Tate theory for moduli spaces of PEL type. Ann. Sci. École Norm. Sup. (4) 37(2), 223–269 (2004)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Moonen, B.: Special subvarieties arising from families of cyclic covers of the projective line. Doc. Math. 15, 793–819 (2010)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Oort, F.: Abelian varieties isogenous to a Jacobian in problems from the Workshop on Automorphisms of Curves. Rend. Sem. Mat. Univ. Padova 113, 129–177 (2005)MathSciNetGoogle Scholar
  27. 27.
    Pries, R.: Current results on Newton polygons of curves. In: Questions in Arithmetic Algebraic Geometry, Advanced Lectures in Mathematics, Chapter 6 (to appear)Google Scholar
  28. 28.
    Pries, R.: The \(p\)-torsion of curves with large \(p\)-rank. Int. J. Number Theory 5(6), 1103–1116 (2009)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Rapoport, M., Richartz, M.: On the classification and specialization of \(F\)-isocrystals with additional structure. Compos. Math. 103(2), 153–181 (1996)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Rapoport, M., Zink, T.: Period Spaces for \(p\)-Divisible Groups, Annals of Mathematics Studies, vol. 141. Princeton University Press, Princeton, NJ (1996)zbMATHGoogle Scholar
  31. 31.
    Shen, X., Zhang, C.: Stratification in good reductions of Shimura varieties of abelian type, preprint, available on arXiv:1707.00439
  32. 32.
    Shimura, G.: Euler products and Eisenstein series. In: CBMS Regional Conference Series in Mathematics, vol. 93, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI (1997)Google Scholar
  33. 33.
    Shimura, G.: An exact mass formula for orthogonal groups. Duke Math. J. 97(1), 1–66 (1999)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Shimura, G.: Some exact formulas on quaternion unitary groups. J. Reine Angew. Math. 509, 67–102 (1999)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Viehmann, E., Wedhorn, T.: Ekedahl-Oort and Newton strata for Shimura varieties of PEL type. Math. Ann. 356(4), 1493–1550 (2013)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Vollaard, I.: The supersingular locus of the Shimura variety for \({\rm GU}(1, s)\). Can. J. Math. 62(3), 668–720 (2010)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Vollaard, I., Wedhorn, T.: The supersingular locus of the Shimura variety of \({\rm GU}(1, n-1)\) II. Invent. Math. 184(3), 591–627 (2011)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Wewers, S.: Construction of Hurwitz spaces, Dissertation, (1998)Google Scholar
  39. 39.
    Xiao, L., Zhu, X.: Cycles on Shimura varieties via geometric Satake. (2017). preprint, available on arXiv:1707.05700
  40. 40.
    Chia-Fu, Y.: Simple mass formulas on Shimura varieties of PEL-type. Forum Math. 22(3), 565–582 (2010)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

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

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