On Maximal Curves in Characteristic Two

The genus g of an Fq2-maximal curve satisfies g = g1 := q(q − 1)/2 or g ≤ g2 := ⌊(q − 1) /4⌊. Previously, Fq2 -maximal curves with g = g1 or g = g2, q odd, have been characterized up to Fq2 -isomorphism. Here it is shown that an Fq2 -maximal curve with genus g2, q even, is Fq2 -isomorphic to the nonsingular model of the plane curve ∑t i=1 y q/2 = x, q = 2, provided that q/2 is a Weierstrass non-gap at some point of the curve. 1. A projective geometrically irreducible nonsingular algebraic curve defined over Fq2, the finite field with q elements, is called Fq2-maximal if the number of its Fq2-rational points attains the Hasse-Weil upper bound q + 1 + 2qg , where g is the genus of the curve. Maximal curves became useful in Coding Theory after Goppa’s paper [Go], and have been intensively studied in [Sti-X], [Geer-Vl1] (see also the references therein), [Geer-Vl2], [FT1], [FGT], [FT2], [GT], [CHKT], [CKT1], [G-Sti-X] and [CKT2]. The key property of a Fq2-maximal curve X is the existence of a base-point-free linear system DX := |(q + 1)P0|, P0 ∈ X (Fq2), defined on X such that [FGT, §1] (1.1) qP + FrX (P ) ∈ DX , (1.2) DX is simple, (1.3) dim(DX ) ≥ 2, where FrX denotes de Frobenius morphism on X relative to Fq2. Then via Stöhr-Voloch’s approach to the Hasse-Weil bound [SV] one can establish arithmetical and geometrical properties of maximal curves. In addition, Property (1.2) allows the use of Castelnuovo’s genus bound in projective spaces [Cas], [ACGH, p. 116], [Ra, Corollary 2.8]. In particular, the following relation involving the genus g of X and n := dim(DX ) − 1 holds [FGT, p. 34] 2g ≤ { (q − n/2)/n if n is even , ((q − n/2) − 1/4)/n otherwise . (1) It follows that g ≤ g1 := q(q − 1)/2 , 1991 Math. Subj. Class.: Primary 11G, Secondary 14G. 1 2 M. ABDÓN AND F. TORRES which is a result pointed out by Ihara [Ih]. As a matter of fact, the so called Hermitian curve, i.e. the plane curve H defined by Y Z + Y Z = X , is the unique Fq2-maximal curve whose genus is g1 up to Fq2-isomorphism [R-Sti]. Moreover, H is the unique Fq2-maximal curve X such that dim(DX ) = 2 [FT2, Thm 2.4]. Therefore, if g < g1, then dim(DX ) ≥ 3 and hence (1) implies [Sti-X], [FT1] g ≤ g2 := ⌊(q − 1) /4⌋ . If q is odd, there is a unique Fq2-maximal curve, up to Fq2-isomorphism, whose genus belongs to the interval ](q− 1)(q− 2)/4, (q− 1)/4], namely the nonsingular model of the plane curve y + y = x , whose genus is g2 = (q − 1) /4 [FGT, Thm. 3.1], [FT2, Prop. 2.5]. The purpose of this paper is to extend this result to even characteristic provided that a condition on Weierstrass non-gaps is satisfied. For q even, say q = 2, notice that g2 = q(q − 2)/4 and that the nonsingular model of the plane curve t