有限域上等变椭圆同态对应的有理函数的置换性与例外性
Indecomposable rational functions over finite fields with Galois closure of genus one: Reconstruction and exceptionality
浏览论文内容
中文总结 AI 辅助
该研究针对有限域上带群作用的椭圆曲线间的等变同态，建立了其诱导有理函数的置换性与例外性的等价判据，明确了相关周期与密度，给出低阶循环商群的显式判据。
中文摘要 AI 辅助
设 $k=\mathbb F_q$，其固定代数闭包为 $\bar k$；设 $E,E'/k$ 为椭圆曲线，具有非平凡有限群 $\Lambda$ 的忠实保原点作用，且 $\varphi:E\to E'$ 为可分的 $\Lambda$-等变 $k$-同态。假设 $E$ 上的指定作用在 $q$-幂 Frobenius 共轭下稳定，令 $f$ 为商线间的诱导映射。我们证明该族无意外置换：对每个 $k_r=\mathbb F_{q^r}$，映射 $f$ 置换 $\mathbf P^1(k_r)$ 当且仅当它在 $k_r$ 上是例外的。更准确地，记 $\mathcal N=\ker\varphi(\bar k)$，这些等价性质完全由 $\mathcal N$ 上的诱导 Frobenius 与 $\Lambda$-作用控制：当且仅当对所有 $\gamma\in\Lambda$，有 $\mathcal N\cap\ker(\mathcal F_E^r-\gamma_E)=\{\mathrm O_E\}$，其中 $\mathrm O_E$ 为 $E$ 的原点，$\mathcal F_E$ 为 $q$-幂 Frobenius，$\gamma_E$ 为 $\gamma$ 在 $E$ 上的作用。加权 Frobenius 区恒等式使该判据在分歧商纤维上仍精确。我们将几何与算术单值群识别为 $\mathcal N$ 上的仿射群，证明置换支撑纯周期，有明确的最小正周期与自然密度；对乘法映射，该判据简化为挠元上的行列式检验，给出阶为2、3、4、6的循环商群的显式分裂与非分裂判据。
英文摘要
Let $k=\mathbb F_q$. We reconstruct every $k$-indecomposable $g\in k(X)$ of degree greater than one whose normal closure has genus one. Such a map is automatically separable and, after independent degree-one changes of the source and target coordinates over $k$, arises from a separable equivariant isogeny between elliptic curves equipped with compatible finite group actions stable under Frobenius conjugation. In particular, $°g=\ell$ or $\ell^2$ for a prime $\ell$, with $\ell\ne\operatorname{char}k$ in the latter case. More generally, every separable rational function with genus-one Galois closure admits a canonical factorization class, modulo degree-one changes of the intermediate coordinates over $k$, determined by the intrinsic translation subgroup of its geometric monodromy group; every $k$-indecomposable factor of the remaining map has Galois closure of genus zero. For every equivariant-isogeny quotient and every finite extension $k_r/k$, the same exact finite-kernel condition characterizes both permutation of $\mathbf P^1(k_r)$ and exceptionality over $k_r$. The finite kernel and its induced Frobenius and linear symmetry actions also determine the arithmetic and geometric monodromy permutation groups, decomposition classes, and the exact periodic set of permutation extension degrees, including its least period and limiting proportion. Together with the corresponding genus-zero theorem, these results yield permutation if and only if exceptionality over every finite extension for every separable rational function whose Galois closure has genus at most one; under $k$-decomposition, the common extension-degree set is the intersection of the corresponding sets for the factors.
发表机构
- School of Mathematics, Sun Yat-sen University（中山大学数学学院）
机构由 AI 辅助整理，请以论文原文为准。
