Slope Estimates of Artin–Schreier Curves

Abstract

Let $X/\overline{\open F}_p$ be an Artin–Schreier curve defined by the affine equation y^p − y = $\tilde{f}$(x) where $\tilde{f}$(x) ∈ $\overline{\open F}_p$[x] is monic of degree d. In this paper we develop a method for estimating the first slope of the Newton polygon of X. Denote this first slope by NP₁($X/\overline{\open F}_p$). We use our method to prove that if p>d ≥ 2 then NP₁($X/\overline{\open F}_p$) ≥ [lceil ](p−1)/d[rceil ]/(p − 1). If p > 2d ≥ 4, we give a sufficient condition for the equality to hold.