Curves over Finite Fields Attaining the Hasse-Weil Upper Bound
نویسنده
چکیده
Curves over finite fields (whose cardinality is a square) attaining the Hasse-Weil upper bound for the number of rational points are called maximal curves. Here we deal with three problems on maximal curves: 1. Determination of the possible genera of maximal curves. 2. Determination of explicit equations for maximal curves. 3. Classification of maximal curves having a fixed genus.
منابع مشابه
On Curves over Finite Fields with Many Rational Points
We study arithmetical and geometrical properties of maximal curves, that is, curves defined over the finite field Fq2 whose number of Fq2 -rational points reachs the Hasse-Weil upper bound. Under a hypothesis on non-gaps at rational points we prove that maximal curves are Fq2 -isomorphic to y q + y = x for some m ∈ Z.
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This bound was proved for elliptic curves by Hasse in 1933. Ever since, the question of the maximum number Nq(g) of points on an irreducible curve of genus g over a finite field of cardinality q could have been investigated. But for a long time it attracted no attention and it was only after Goppa introduced geometric codes in 1980 that this question aroused systematic attention, cf. [G], [M], ...
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تاریخ انتشار 2000