具有$\mathbb{Q}(ζ_p,\sqrt[p]{q})$形式模域正则设计
Regular dessins with moduli fields of the form $\mathbb{Q}(ζ_p,\sqrt[p]{q})$
摘要：
Gareth Jones在2014年SIGMAP会议上询问了具有非交换模场的正则dessins的例子。本文首先构造了模域为$\mathbb{Q}(ζ_p,\sqrt[p]{q})$形式的非交换Galois扩张的dessins，其中$p$是奇素数，$ζ_p$是$p$的单位根，$q\in\mathbb{Q}$不是$p$的次幂，然后证明了它们的正则闭包具有相同的模域.最后，在$p=q=3$的特殊情况下，给出了模域$\mathbb{Q}(ζ_3,\sqrt[3]{3})$的度为$2^{19}\cdot3^4$亏格为$14155777$的正则设计的另一个例子.
英文摘要：
Gareth Jones asked during the 2014 SIGMAP conference for examples of regular dessins with nonabelian fields of moduli. In this paper, we first construct dessins whose moduli fields are nonabelian Galois extensions of the form $\mathbb{Q}(ζ_p,\sqrt[p]{q})$, where $p$ is an odd prime and $ζ_p$ is a $p$th root of unity and $q\in\mathbb{Q}$ is not a $p$th power, and we then show that their regular closures have the same moduli fields. Finally, in the special case $p=q=3$ we give another example of a regular dessin with moduli field $\mathbb{Q}(ζ_3,\sqrt[3]{3})$ of degree $2^{19}\cdot3^4$ and genus $14155777$.
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