On maximal curves

Details On maximal curves On maximal curves

1996-10-31

arxiv.org/abs/alg-geom/9610023v1

Mathematics

Algebraic Geometry

LaTex2e, 17 pages; this article is an improved version of the paper alg-geom/9603013 (by Fuhrmann and Torres)

Scientific paper

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 $m \in Z^+$. 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}.

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