On plane maximal curves

The number N of rational points on an algebraic curve of genus g over a fin ite field bb F-q satisfies the Hasse-Weil bound N less than or equal to q 1 +2g root q. A curve that attains this bound is called maximal. With g(0) =1/2(q - root q) and g(1) = 1/4(root q- 1)(2), it is known that maximalcur ves have g = g(0) or g less than or equal to 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) = 1/8(root q - 1)(root q - 3), and for this genus there are two non-isomorphic maximal curves known when root q = 3 (mod 4). Here, a maximal curve with genus g(2) and a non-singular plane model is character ized as a Fermat curve of degree 1/2(root q + 1).