The genus of curves over finite fields with many rational points

We prove the following result which was conjectured by Stichtenoth and Xing: letg be the genus of a projective, irreducible non-singular algebraic curve over the finite field and whose number of -rational points attains the Hasse-Weil bound; then either 4g≤(q−1)² or 2g=(q−1)q. Supported by a grant from the International Atomic Energy and UNESCOCorrespondence to: F. Torres This article was processed by the author using theLatex style file from Springer-Verlag.