AI 中文总结
对特征不为2和5的有限域上次数为5的例外有理函数进行分类，利用算术和几何单值群、分歧结构及黎曼 - 赫维茨公式，给出循环和二面体单值性情况的结果，还得到特征2和5的部分分类。
AI 中文摘要
我们完成了在特征不同于2和5的任意有限域$\mathbb{F}_{q}$上次数为5的例外有理函数（直至莫比乌斯等价）的分类。我们的方法采用算术和几何单值群，结合相关覆盖的分歧结构和黎曼 - 赫维茨公式。循环单值性情况恰好产生单项式和雷代伊函数，而二面体情况通过分析其惯性群和分支点来解决；这导致迪克森多项式和一个由椭圆曲线的有理5 - 同构产生的非多项式族。我们还在特征2和5中获得了部分分类：确定了所有具有循环几何单值性的情况以及所有允许$\mathbb{F}_{q}$ - 有理分支点的二面体情况。唯一未解决的情况是那些几何单值群为$D_{5}$且没有$\mathbb{F}_{q}$ - 有理分支点的情况。
英文摘要
We give a complete classification, up to $k$-Möbius equivalence, of exceptional rational functions of degree five over every finite field. The classification gives explicit normal forms in every characteristic, exact parameter identifications, and the resulting class counts. The main structural input is an arithmetic theory of shifted dihedral isogeny quotients valid in every odd degree. For each odd $n\geqslant3$, the separable degree-$n$ rational maps whose geometric monodromy group is isomorphic to $D_n$ and whose nontrivial inertia groups are generated by reflections are precisely the maps induced by cyclic $n$-isogenies on shifted Kummer quotients. We determine the exact ambiguity of the isogeny data under two-sided $k$-Möbius equivalence by means of a signed Frobenius-descent invariant. The theory allows arbitrary odd $n$, reduced kernels when the characteristic divides $n$, and wild reflection inertia. If Frobenius acts on the cyclic kernel by $λ\in(\mathbb Z/n\mathbb Z)^\times$, the induced map is exceptional exactly when both $λ-1$ and $λ+1$ are units modulo $n$.
Comments41 pages. In v3, we add exact $k$-Möbius equivalence criteria and class counts, and develop an arithmetic theory of shifted dihedral isogeny quotients in every odd degree
