论文标题
在$ p $ - 基于定义的字段
On $p$-gonal fields of definition
论文作者
论文摘要
让$ s $是$ g \ geq 2 $的封闭式riemann表面，$φ$是Prime Order $ p $ $ s $ p $ $ s $ p $的保形自动形态，因此$ s/\langleφ\ rangle $具有零属。令$ {\ mathbb k} \ leq {\ mathbb c} $为$ s $的定义字段。我们为存在$ {\ mathbb k} $的字段扩展的存在提供了一个论点，最多是$ 2（p-1）$，为此，$ s $可以通过$ y^{p} = f（p} = f（p} = f（x）\ in {\ nathbb f} $的曲线的曲线可以定义（x，e^{2πi/p} y）$。此外，如果在$ {\ mathbb k} $上也可以定义$φ$，则可以选择$ {\ mathbb f} $最多最多是$ {\ mathbb k} $的二次扩展。对于$ p = 2 $，那就是$ s $是过度椭圆形的，而$φ$是其过度递减的互动，这是由于Mestre（甚至是属）和Huggins和Lercier-ritzenthaler-Sijslingit所致，如果$ {\ rm aut}（\ rm aut}（s）/φ\ rangle $是非trivial的情况。
Let $S$ be a closed Riemann surface of genus $g \geq 2$ and $φ$ be a conformal automorphism of $S$ of prime order $p$ such that $S/\langle φ\rangle$ has genus zero. Let ${\mathbb K} \leq {\mathbb C}$ be a field of definition of $S$. We provide an argument for the existence of a field extension ${\mathbb F}$ of ${\mathbb K}$, of degree at most $2(p-1)$, for which $S$ is definable by a curve of the form $y^{p}=F(x) \in {\mathbb F}[x]$, in which case $φ$ corresponds to $(x,y) \mapsto (x,e^{2 πi/p} y)$. If, moreover, $φ$ is also definable over ${\mathbb K}$, then ${\mathbb F}$ can be chosen to be at most a quadratic extension of ${\mathbb K}$. For $p=2$, that is when $S$ is hyperelliptic and $φ$ is its hyperelliptic involution, this fact is due to Mestre (for even genus) and Huggins and Lercier-Ritzenthaler-Sijslingit in the case that ${\rm Aut}(S)/φ\rangle$ is non-trivial.
