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

TL;DR

This paper links the first slope of Artin-Schreier curves to p-adic weights of polynomial support and constructs curves with specific slopes using p-symmetry conditions.

Contribution

It introduces a p-adic combinatorial condition called p-symmetry and constructs explicit curves with prescribed first slopes in characteristic p.

Findings

The first slope's lower bound is 1/s_p(v) under p-symmetry.

Unique maximal p-adic weight element v determines slope achievement.

Explicit families of curves with first slope 1/n are constructed for all p.

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.