Newton polygons of cyclic covers of the projective line branched at three points

2018/05/11 by Wanlin Li, Elena Mantovan, Li, Wanlin +5
Computer Science · Engineering · Mathematics · #11G10 #11G20 #11M38 #14G10 #14H10 #14H30 #14H40 #14H40 (secondary) #14K22 (primary) #Advanced Numerical Analysis Techniques #Algebraic Geometry and Number Theory #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.1805.04598

openalex publication_date 2018/05/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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