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On Plane Maximal Curves

Authors:	A. Cossidente J. W. P. Hirschfeld G. Korchmáros F. Torres

Affiliation:	(1) Dipartimento de Matematica, Università della Basilicata, Potenza, 85100, Italy;(2) School of Mathematical Sciences, University of Sussex, Brighton, BN1 9QH, United Kingdom;(3) IMECC-UNICAMP, Cx. P. 6065, Campinas, 13083-970-SP, Brazil

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 . A curve that attains this bound is called maximal. With $g_0 = frac{1}{2}(q - sqrt q )$ and $g_1 = frac{1}{4}(sqrt q - 1)^2$ , it is known that maximalcurves have $g = g_0 or g leqslant {text{ }}g_1$ . Maximal curves with have been characterized up to isomorphism. A natural genus to be studied is $g_2 = frac{1}{8}(sqrt q - 1)(sqrt q - 3),$ and for this genus there are two non-isomorphic maximal curves known when . Here, a maximal curve with genus g₂ and a non-singular plane model is characterized as a Fermat curve of degree $frac{1}{2}(sqrt q + 1)$ .

Keywords:	finite field maximal curve linear series
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