标题： p-秩为零的代数曲线的自同构群 显示英文标题

标题： Automorphism groups of algebraic curves with p-rank zero

摘要： 在正特征的情况下，代数曲线的自同构群可以比经典的Hurwitz界预期的多得多。 甚至存在任意高亏格 \$g\$ 的代数曲线，其自同构数量超过 \$16g^4\$。 人们多次观察到，最异常的例子无一例外地具有零 \$p\$-秩。 本文研究了定义在特征为 2 的代数闭域 \$K\$ 上的零 \$2\$-秩代数曲线 \$X\$ 的 \$K\$-自同构群 \$\mathrm\{Aut\}(X)\$。 主要结果是，如果曲线的亏格 \$g \geq 2\$，且 \$|\mathrm\{Aut\}(X)| > 24g^2\$，那么除了少数例外情况外，\$\mathrm\{Aut\}(X)\$ 在 \$X\$ 上有一个不动点。 在例外情况下，确定了 \$\mathrm\{Aut\}(X)\$ 和 \$g\$ 的可能性。 显示英文摘要

摘要： In positive characteristic, algebraic curves can have many more automorphisms than expected from the classical Hurwitz's bound. There even exist algebraic curves of arbitrary high genus g with more than 16g^4 automorphisms. It has been observed on many occasions that the most anomalous examples invariably have zero p-rank. In this paper, the K-automorphism group Aut(X) of a zero 2-rank algebraic curve X defined over an algebraically closed field K of characteristic 2 is investigated. The main result is that if the curve has genus g greater than or equal to 2, and |Aut(X)|>24g^2, then Aut(X) has a fixed point on X, apart from few exceptions. In the exceptional cases the possibilities for Aut(X) and g are determined.