Construction of Curves with a Controlled First Slope using p-Symmetric Numbers

2024/11/04 by Robert Moore, Hui June Zhu, Moore, Robert +1
Mathematics · #Advanced Mathematical Identities #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Number Theory (math.NT) #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2411.01832

openalex publication_date 2024/11/04 · openalex created_date 2024/11/14 · openalex updated_date 2026/07/28

Abstract

This paper establishes a constructive link between the first slope of Artin-Schreier curves X_f: y^p-y=f(x) and the p-adic weight of the support of f(x). If the maximal p-adic weight element v in Supp(f) is unique, we show that the first slope's lower bound of 1/s_p(v) is achieved if and only if v satisfies a combinatorial p-adic condition, which we define as p-symmetry. As an application, we construct explicit families of curves in every characteristic p with first slope equal to 1/n for every n>2.

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