On curves over finite fields with many rational points

Abstract

We study arithmetical and geometrical properties of {\it maximal curves}, that is, curves defined over the finite field ${\mathbb{F}}_{q^2}$ whose number of ${\mathbb{F}}_{q^2}$-rational points reachs the Hasse-Weil upper bound. Under a hypothesis on non-gaps at rational points we prove that maximal curves are ${\mathbb{F}}_{q^2}$-isomorphic to $y^q+y=x^m$ for some $m\in {\mathbb{Z}}^{+}$.

LaTex2e, 10 pages