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Further examples of maximal curves. (English) Zbl 1166.11017
A smooth, projective and geometrically irreducible curve \(C\) of genus \(g\) over a finite field \(\mathbb F_{q^{2}}\) is called maximal over \(\mathbb F_{q^{2}}\) if \(\# C(\mathbb F_{q^{2}})\) attains the Hasse-Weil upper bound, i.e., \(\# C(\mathbb F_{q^{2}})=q^{2}+1+2gq\). For such curves, one necessarily has \(g\leq q(q-1)/2\). Further, there exists a unique maximal curve \(\mathcal H\) over \(\mathbb F_{q^{2}}\), with affine model \(y^{q}+y=x^{q+1}\) and called the Hermitian curve, such that \(g\) has the maximal value \(q(q-1)/2\).
This paper studies some classes of maximal curves and discusses whether or not the curves studied are Galois-covered by \(\mathcal H\). First, the authors study the nonsingular models of the curves \(C(q,n): y^{q^{2}}-y=x^{\frac{q^{n}+1}{q+1}}\) defined over \(\mathbb F_{q^{2n}}\), where \(n\geq 3\) is an odd integer. If \(\ell\) is a prime, the \(\mathbb F_{\ell^{6}}\)-curves \(C(\ell,3)\) were previously studied by A. García and H. Stichtenoth. They showed that these curves are maximal over \(\mathbb F_{\ell^{6}}\) of genus \((\ell^{2}-1)(\ell^{2}-\ell)/2\). An interesting observation is the following: the curve \(C(3,3)\) is not Galois-covered by the Hermitian curve \(\mathcal H\), in contrast to the case of \(C(2,3)\).
In this paper the curves \(C(q,n)\) are shown to be maximal for any \(q\). Further, it is shown that the curves \(C(2,n)\) are Galois-covered by \(\mathcal H\). The authors also discuss the family of maximal curves \(C_{b}\) defined over \(\mathbb F_{q^{2n}}\) by the affine equation \(y^{N}=-x^{b}(x+1), 1\leq b\leq N-1\), where \(N\) is an odd divisor of \(q^{n}+1\) and \((N,b)=(N,b+1)=1\).

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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