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The corresponding Hasse-Weil bound was a major breakthrough in history ofmathematics. It has found many applications in mathematics, coding theory andtheoretical computer science. In general, the Hasse-Weil bound is tight andcannot be improved. However, the Hasse-Weil bound is no longer tight when it isapplied to some specific classes of curves. One of the examples where theHasse-Weil bound is not tight is the family of Artin-Schreier curves. Due tovarious applications of Artin-Schreier curves to coding, cryptography andtheoretical computer science, researchers have made great effort to improve theHasse-Weil bound for Artin-Schreier curves.  In this paper, we focus on the number of rational places of theArtin-Schreier curve defined by $y^p-y=f(x)$ over the finite field$\mathbb{F}_q$ of characteristic $p$, where $f(x)$ is a polynomial in$\mathbb{F}_q[x]$. Our road map for attacking this problem works as follows. Wefirst show that the function field $E_f:=\mathbb{F}_q(x,y)$ of theArtin-Schreier curve $y^p-y=f(x)$ is a subfield of some cyclotomic functionfield. We then make use of the class field theory to prove that the number ofpoints of the curve is upper bounded by a function of a minimum distance of alinear code. By analyzing the minimum distance of this linear code, we canimprove the Hasse-Weil bound and Serre bound for Artin-Schreier curves.

An improvement of the Hasse-Weil bound for Artin-Schreier curves via cyclotomic function fields