2001/05/01 by Jasper Scholten, Hui June Zhu
Mathematics · #math.AG #math.NT
published as Compositio Math. 137 (2003), 275-292. · 20 pages, Latex
arxiv created 2001/05/01 · arxiv updated 2009/11/30
Let f(x) = xd + ad-1xd-1 + ... + a0 be a polynomial of degree d in Q[x]. For every prime number p coprime to d and f(x) in (Zp ∩ Q)[x], let X/Fp be the Artin-Schreier curve defined by the affine equation yp - y = f(x) mod p. Let NP1(X/Fp) be the first slope of the Newton polygon of X/Fp. We prove that there is a Zariski dense subset U in the space Ad of degree-d monic polynomials over Q such that for all f(x) in U, the limit of NP1(X/Fp) is equal to 1/d as p goes to infinity. This is a ``first slope version'' of a conjecture of Wan. Let X/Fpbar be an Artin-Schreier curve defined by the affine equation yp - y = F(x) where F(x) = xd + Ad-1xd-1 + ... + A0. We prove that if p>d>1 then NP1(X/Fpbar) >= ceiling((p-1)/d)/(p-1). If p>2d>3, we give a sufficient condition for the equality to hold.