Divisibility of L-Polynomials for a Family of Artin-Schreier Curves

In this paper we consider the curves \(C_k^{(p,a)} : y^p-y=x^{p^k+1}+ax\) defined over \(\mathbb F_p\) and give a positive answer to a conjecture about a divisibility condition on \(L\)-polynomials of the curves \(C_k^{(p,a)}\). Our proof involves finding an exact formula for the number of \(\mathbb F_{p^n}\)-rational points on \(C_k^{(p,a)}\) for all \(n\), and uses a result we proved elsewhere about the number of rational points on supersingular curves.