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

Abstract

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.