Slope Estimates of Artin-schreier Curves

Let X/Fp be an Artin-Schreier curve defined by the affine equation yp−y = f̃(x) where f̃(x) ∈ Fp[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/Fp)). We use our method to prove that if p > d ≥ 2 then NP1(X/Fp) ≥ d p−1 d e p−1 . If p > 2d ≥ 4, we give a sufficient condition for the equality to hold.