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

TL;DR

This paper proves a divisibility conjecture for L-polynomials of a family of Artin-Schreier curves by deriving explicit formulas for their rational points over finite fields, leveraging properties of supersingular curves.

Contribution

It provides a proof of a divisibility conjecture for L-polynomials of specific Artin-Schreier curves, using explicit point count formulas and supersingular curve properties.

Findings

Confirmed divisibility of L-polynomials for the curves

Derived explicit formulas for point counts over finite fields

Connected properties of supersingular curves to L-polynomial divisibility

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.