AI 中文总结
本文对几何伽罗瓦闭包亏格1的可分不可约驯服例外射影直线映射分类，给出有限域上算术固定域形式的编码准则，获秩2支撑公式等结果，明确特征与障碍的相关性质。
AI 中文摘要
设k=𝔽_q，我们在双侧k-Möbius等价意义下，对所有几何伽罗瓦闭包具有亏格1的可分不可约驯服例外映射P¹_k→P¹_k进行分类。所得的算术固定域形式由覆盖导出的Frobenius稳定仿射椭圆商数据编码；我们证明了一个逆命题和一个精确等价准则。相同数据决定了分支算术、几何与算术单值群、常数域以及在每个有限扩域上的行为。特别地，在每个𝔽_{q^r}上，置换等价于例外性。我们得到了显式的秩2支撑公式、必要充分出现准则，以及每个驯服符号下的精确双侧类计数，包括特征2和3下的存活情形。在特征大于3时，驯服性是自动的。仅在更强的三次自同态实现问题中会出现尖锐的3-adic障碍，而非固定域分类本身。
英文摘要
Let $k=\mathbb F_q$. We classify, up to two-sided $k$-Möbius equivalence, all separable indecomposable tame exceptional maps $\mathbf P^1_k\to\mathbf P^1_k$ whose geometric Galois closure has genus one. The resulting arithmetic fixed-field forms are encoded by Frobenius-stable affine elliptic quotient data recovered from the cover; we prove a converse and an exact equivalence criterion. The same data determine branch arithmetic, geometric and arithmetic monodromy, the constant field, and behavior over every finite extension. In particular, over each $\mathbb F_{q^r}$, permutation is equivalent to exceptionality. We obtain explicit rank-two support formulas, necessary and sufficient occurrence criteria, and exact two-sided class counts in every tame signature, including the surviving cases in characteristics $2$ and $3$. In characteristic greater than $3$, tameness is automatic. A sharp $3$-adic obstruction arises only for a stronger cubic self-endomorphism realization problem, not for the fixed-field classification itself.
Comments64 pages
