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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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