ar X iv : a lg - g eo m / 9 61 00 23 v 1 3 1 O ct 1 99 6 ON MAXIMAL CURVES

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 reaches the Hasse-Weil upper bound. Under a hypothesis on non-gaps at a rational point, we prove that maximal curves are F q 2-isomorphic to y q + y = x m , for some m ∈ Z +. As a consequence we show that a maximal curve of genus g = (q − 1) 2 /4 is F q 2-isomorphic to the curve y q + y = x (q+1)/2. The interest on curves over finite fields was renewed after Goppa [Go] showed their applications to Coding Theory. One of the main features of linear codes arising from curves is the fact that one can state a lower bound for their minimum distance. This lower bound is meaningful only if the curve has many rational points. The subject of this paper is the study of maximal curves. Let X be a projective, geometrically irreducible and non-singular algebraic curve defined over the finite field F ℓ with ℓ elements. A celebrated theorem of Weil states that: # X(F ℓ) ≤ ℓ + 1 + 2g √ ℓ, where X(F ℓ) denotes the set of F ℓ-rational points of X and g is the genus of the curve. This bound was proved for elliptic curves by Hasse. The curve X is called maximal over F ℓ (in this case, ℓ must be a square; say ℓ = q 2) if it attains the Hasse-Weil upper bound; that is, # X(F q 2) = q 2 + 1 + 2gq. Ihara [Ih] shows that the genus of a maximal curve over F q 2 satisfies: g ≤ (q − 1)q/2. Rück and Stichtenoth [R-Sti] show that the Hermitian curve (that is, the curve given by y q + y = x q+1) is the unique (up to F q 2-isomorphisms) maximal curve over F q 2 having genus g = (q − 1)q/2. It is also known that the genus of maximal curves over F q 2 satisfies (see [F-T] and the remark after Theorem 1.4 here): g ≤ (q − 1) 2 /4 or g = (q − 1)q/2. The Hermitian curve is a particular case of the following maximal curves over F q 2 : y q + y = x m , with …