Divisibility of L-polynomials for a family of Artin–Schreier curves

Abstract

In this paper we consider the curves Ck(p,a):yp−y=xpk+1+ax defined over Fp and give a positive answer to a conjecture about a divisibility condition on L-polynomials of the curves Ck(p,a). Our proof involves finding an exact formula for the number of Fpn-rational points on Ck(p,a) for all n, and uses a result we proved elsewhere about the number of rational points on supersingular curves.