关于特征$p>0$具有大自同构$p$ -群的平滑曲线
On smooth curves endowed with a large automorphism $p$-group in characteristic $p>0$
摘要：
设$k$是特征$p>0$的代数闭域，$C$是$k$上亏格为$g \geq 2$的连通非奇异射影曲线.本文继续了Lehr和Matignon开始的工作，即研究“大行动”，即对$(C,G)$，其中$G$是$p$ -子群的$k$ -自同构群的$C$，使$\frac{|G|}{g} >\frac{2 p}{p-1}$。如果$G_2$表示$G$在覆盖$C \to C/G$的唯一分歧点处的第二分歧群，则我们给出了$(C,G)$是大行动的$G_2$上的必要条件，这允许我们追求大行动的分类。我们的例子主要来源于使用全局函数场的射线类场论来构造具有许多有理点的曲线，这是由J-P. Serre发起的，随后是Lauter和Auer。特别是，我们得到明确的例子，大行动与$G_2$阿贝尔指数大。
英文摘要：
Let $k$ be an algebraically closed field of characteristic $p>0$ and $C$ a connected nonsingular projective curve over $k$ with genus $g \geq 2$. This paper continues the work begun by Lehr and Matignon, namely the study of "big actions", i.e. the pairs $(C,G)$ where $G$ is a $p$-subgroup of the $k$-automorphism group of $C$ such that$\frac{|G|}{g} >\frac{2 p}{p-1}$. If $G_2$ denotes the second ramification group of $G$ at the unique ramification point of the cover $C \to C/G$, we display necessary conditions on $G_2$ for $(C,G)$ to be a big action, which allows us to pursue the classification of big actions. Our main source of examples comes from the construction of curves with many rational points using ray class field theory for global function fields, as initiated by J-P. Serre and followed by Lauter and Auer. In particular, we obtain explicit examples of big actions with $G_2$ abelian of large exponent.
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