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On \(F_{q^2}\)-maximal curves of genus \(\frac{1}{6}(q-3)q\). (English) Zbl 1073.11043
Let \(q=3^t\), where \(t\) is a positive integer. The authors study maximal curves of genus \((q-3)q/6\) defined over the finite field \(K={\mathbb F}_{q^2}\). They show that when \(t\geq 3\), the nonsingular model of the plane curve defined by the equation \(\sum_{i=0}^{t-1} y^{3^i}=ax^{q+1}\), where \(a\in K\) such that \(a^{q-1}=-1\), is such a maximal curve. The authors show that there may be only one other possibility for such a maximal curve of this genus – a nonreflexive space curve of degree \(q+1\) whose tangent surface is also nonreflexive. However, they are not able to determine whether such a nonreflexive curve actually exists. The proof uses results of R. D. M. Accola [Trans. Am. Math. Soc. 251, 357–373 (1979; Zbl 0417.14021)], as well as the Castelnuovo bound and Stöhr-Voloch Weierstrass point theory. These techniques have been used in earlier papers by the second author and G. Korchmáros [for example, Math. Ann. 323, No. 3, 589–608 (2002; Zbl 1018.11029)].

MSC:
11G20 Curves over finite and local fields
14G05 Rational points
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
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