论文标题
$ \ mathbb {p}^1的内态的自动形态组（\ bar {\ mathbb {f}} _ p）$
Automorphism Groups of Endomorphisms of $\mathbb{P}^1 (\bar{\mathbb{F}}_p)$
论文作者
论文摘要
对于任何代数封闭的字段$ k $和任何内态$ f $ $ \ mathbb {p}^1（k）$至少2个学位，$ f $的自动形态是$ f $通勤的Möbius变换，这些转换与$ f $，并且这些形成了$ \ operatorname的有限亚组$ \ opertotorname {pgl} $ _2（k）。在复杂动力学系统的模量空间中，已经详细研究了具有非平凡自动形态的地图的轨迹，并且有一些技术可以构建具有规定的自动形态组的地图，这些图可以追溯到Klein。当$ k $是代数关闭$ \ bar {\ mathbb {f}} _ p $时，我们研究相应的问题。我们使用$ \ operatorname {pgl} _2（\ bar {\ mathbb {f}} _ p）$的有限子组的分类，以表明每个有限子组都可以作为自动形态组实现。为了构建示例，我们使用模块化不变理论的方法。然后，我们计算出$ \ bar {\ mathbb {f}} _ p $ $ 2 $的地图的基因座，而非平凡的自动形态，显示了几何和可能的自动形态群体如何取决于prime $ p $。
For any algebraically closed field $K$ and any endomorphism $f$ of $\mathbb{P}^1(K)$ of degree at least 2, the automorphisms of $f$ are the Möbius transformations that commute with $f$, and these form a finite subgroup of $\operatorname{PGL}_2(K)$. In the moduli space of complex dynamical systems, the locus of maps with nontrivial automorphisms has been studied in detail and there are techniques for constructing maps with prescribed automorphism groups that date back to Klein. We study the corresponding questions when $K$ is the algebraic closure $\bar{\mathbb{F}}_p$ of a finite field. We use the classification of finite subgroups of $\operatorname{PGL}_2(\bar{\mathbb{F}}_p)$ to show that every finite subgroup is realizable as an automorphism group. To construct examples, we use methods from modular invariant theory. Then, we calculate the locus of maps over $\bar{\mathbb{F}}_p$ of degree $2$ with nontrivial automorphisms, showing how the geometry and possible automorphism groups depend on the prime $p$.
