发表机构
University of Science and Technology of China（中国科学技术大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文为Hermitian函数域关于分解群的每个子群构造Galois子域的显式生成元和绝对不可约定义方程，并利用$q=27$时的方程首次给出被同一Hermitian函数域覆盖但非Galois覆盖的极大函数域的例子。
AI 中文摘要
设$q$为素数幂，$\mathbb{F}_{q^2}$为含$q^2$个元素的有限域。由$y^q+y=x^{q+1}$定义的Hermitian函数域$H=\mathbb{F}_{q^2}(x,y)$是著名的极大函数域，具有可能的最大亏格。设$A(P_\infty)$为$H$的无穷远点$P_\infty$的分解群，该点是$x$和$y$的公共极点。对于$A(P_\infty)$的每个子群$G$，我们构造$H$关于$G$的Galois子域$H^G$的显式生成元，并确定定义该Galois子域的光滑仿射平面模型的一个绝对不可约方程。对于$p$-子群，生成元可以选择使得定义方程左边为加性多项式，右边为$\mathbb{F}_p$-二次多项式。对于$q=27$，我们可以从Hermitian函数域的Galois子域的显式方程构造一个亏格为二的子域$D\subset H$，该子域不同构于$H^J$（对$\text{Aut}(H)$的任何子群$J$）。据我们所知，这是第一个被同一Hermitian函数域覆盖但非Galois覆盖的极大函数域的例子。
英文摘要
Let $q$ be a prime power and $\mathbb{F}_{q^2}$ be the finite fields of $q^2$ elements. The Hermitian function field $H=\mathbb{F}_{q^2}(x,y)$ defined by $y^q+y=x^{q+1}$ is a well-known maximal function field with the largest possible genus. Let $A(P_\infty)$ be the decomposition group of the infinity place $P_\infty$ of $H$ which is the common pole of $x$ and $y$. For every subgroup $G\le A(P_\infty)$, we construct explicit generators of Galois subfield $H^G$ of $H$ with respect to $G$ and determine an absolutely irreducible equation defining the smooth affine plane model for such a Galois subfield. For $p$-subgroups, the generators can be chosen so that the defining equation has an additive polynomial on the left-hand side and an $\mathbb{F}_p$-quadratic polynomial on the right-hand side. For $q=27$, we can construct a genus-two subfield $D\subset H$ that is not isomorphic to $H^J$ for any subgroup $J\le \text{Aut}(H)$ from the explicit equations of Galois subfields of the Hermitian function field. To the best of our knowledge, this is the first example of a maximal function field covered but not Galois-covered by the same Hermitian function field.
