Abstract. We study arithmetical and geometrical properties of maximal curves, that is, curves defined over the finite field F q 2 whose number of F q 2 -rational points reachs the Hasse-Weil upper bound. Under a hypothesis on non-gaps at rational points we prove that maximal curves are F q 2 -isomorphic to y q + y = x m for some m ∈ Z + . IntroductionGoppa in [Go] showed how to construct linear codes from curves defined over finite fields. One of the main features of these codes is the fact that one can state a lower bound for the minimum distance of the codes. In fact, let C X (D, G) be a Goppa code defined over a curve X over the finite field F q with q elements, whereCertainly this bound is meaningful only if n is large enough. This provides motivation for the study of curves over finite fields with many rational points.The purpose of this paper is to study maximal curves, that is, curves X over F q whose number of rational points #X(F q ) reaches the Hasse-Weil upper bound. In this case one knows that q must be a square.Let k be the finite field with q 2 elements, where q is a power of a prime p. Let X be a projective, connected, non-singular algebraic curve defined over k which is maximal, that is, #X(k) satisfies #X(k) = q 2 + 2gq + 1. (0.1) Let P ∈ X(k) and set D = g n+1 q+1 the k-linear system on X defined by the divisor (q+1)P . Then n ≥ 1, and D is independent of P . In fact D is a simple base-point-free linear system on X (Corollary 1.2.3, Remark 1.2.5 (ii)). This allow us to apply Stöhr-Voloch's approach concerning Weierstrass point theory over finite fields [S-V]. Moreover, the dimension n+1 of D and the genus g are related by Castelnuovo's genus bound for curves in projective spaces ([C], [ACGH, p.116], [Ra, Corollary 2.8 ]).It is known that 2g ≤ (q − 1)q ([Sti, V.3.3]), and that the Hermitian curve is the unique maximal curve whose genus is (q − 1)q/2 [R-Sti]. Furthermore in [F-T] we proved the following stronger bounds for the genus, namely 4g ≤ (q − 1) 2 or 2g = (q − 1)q.
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