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On a characterization of certain maximal curves. (English) Zbl 1046.11045
Let $$X$$ denote a maximal curve over the field $${\mathbb F}_{q^2}$$, and let $$p$$ denote the characteristic of this field. If there exists a rational point $$P$$ on $$X$$ and a function $$y\in {\mathbb F}_{q^2}(X)$$ with $$(y)_\infty=mP$$, where $$m| (q+1)$$, then R. Fuhrmann, F. Torres, and the second author [J. Number Theory 67, 29–51 (1997; Zbl 0914.11036)] showed that $$X$$ must be isomorphic to the nonsingular model of $$y^q+y=x^m$$. If, instead, $$m| q$$ and $$x\in {\mathbb F}_{q^2}(X)$$ with $$(x)_\infty=mP$$, then they conjectured that $$X$$ is isomorphic to the curve given by $$P(z)=x^{q+1}$$, where $$P(z)$$ is an $${\mathbb F}_p$$-linear polynomial of degree $$m$$.
The authors prove a weaker result: Under the additional hypothesis that the field $${\mathbb F}_{q^2}(X)$$ is a Galois extension of $${\mathbb F}_{q^2}(x)$$, they show that in this case the curve is isomorphic to a curve given by $$P(z)=A(x)$$, where $$P(z)\in {\mathbb F}_{q^2}[z]$$ is an additive separable polynomial of degree $$m$$ and $$A(x)\in {\mathbb F}_{q^2}[x]$$ is a polynomial of degree $$q+1$$.
In the particular case $$m=q/p$$, they show, without the Galois assumption above, that $$X$$ is isomorphic to the nonsingular model of the curve given by $\sum_{i=1}^t z^{p^{t-i}}=c\cdot x^{q+1},$ where $$q=p^t$$ and $$c^{q-1}+1=0$$. This generalizes a result in characteristic two shown by the first author and F. Torres [Manuscr. Math. 99, 39–53 (1999; Zbl 0931.11022)]. When $$m$$ is a proper divisor of $$q$$ such that $$m<q/p$$, the authors give several examples to demonstrate nonuniqueness.

##### MSC:
 11G20 Curves over finite and local fields 14G15 Finite ground fields in algebraic geometry 14H25 Arithmetic ground fields for curves
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##### References:
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