The genus of curves over finite fields with many rational points

AbstractWe 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 $$\mathbb{F}_{q^2 } $$ and whose number of $$\mathbb{F}_{q^2 } $$ -rational points attains the Hasse-Weil bound; then either 4g≤(q−1)2 or 2g=(q−1)q.