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)


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.


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


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


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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

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