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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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