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

Wanlin Li, Elena Mantovan, Rachel Pries, Yunqing Tang

Research output: Chapter in Book/Report/Conference proceedingChapter

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.

Original languageEnglish (US)
Title of host publicationAssociation for Women in Mathematics Series
PublisherSpringer
Pages115-132
Number of pages18
DOIs
StatePublished - 2019

Publication series

NameAssociation for Women in Mathematics Series
Volume19
ISSN (Print)2364-5733
ISSN (Electronic)2364-5741

All Science Journal Classification (ASJC) codes

  • Mathematics(all)
  • Gender Studies

Keywords

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

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