On Plane Maximal Curves

Abstract

The number N of rational points on an algebraic curve of genus g over a finite field ${\mathbb F}_q$ satisfies the Hasse–Weil bound $N \leq q + 1 +2g\sqrt {q}$. A curve that attains this bound is called maximal. With $g_0 =\textstyle {1\over 2}(q -\sqrt {q})$ and $g_1= \textstyle {1\over 4}(\sqrt {q}- 1)^2$, it is known that maximalcurves have $g= g_0$ or $g \leq g_1$. Maximal curves with $g=g_0$ or $g_1$ have been characterized up to isomorphism. A natural genus to be studied is $g_2 =\textstyle {1 \over 8}(\sqrt {q}- 1)(\sqrt {q}- 3),$ and for this genus there are two non-isomorphic maximal curves known when $\sqrt {q}\equiv 3\;({\rm mod}\; {4})$. Here, a maximal curve with genus $g_2$ and a non-singular plane model is characterized as a Fermat curve of degree $ {1\over 2}(\sqrt {q} + 1)$.