迭代嘉当通量与有限域上驯服多项式自同构群的非有限生成性
Iterated Cartier Flux and Non-Finite Generation of Tame Polynomial Automorphism Groups over Finite Fields
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中文总结 AI 辅助
该研究针对有限域上的驯服多项式自同构群，通过迭代嘉当通量的性质证明其仅在一维时有限生成，进而证实了 $n\ge3$ 时的 Maubach–Willems 有限生成猜想。
中文摘要 AI 辅助
设 $k=\mathbb{F}_q$ 为特征 $p$ 的有限域，令 $U_n(k)=\{F\in\operatorname{TA}_n(k):F(0)=0,\\ JF(0)=I_n\}$。我们证明，每个特殊驯服多项式自同构对与 $\lambda=x_1\\,dx_2\wedge\cdots\wedge dx_n$ 相关的 de Rham 通量类均为重复嘉当可容许的。迭代嘉当下降的最低权投影给出加法特征 $\chi_{r,j}:U_n(k)\longrightarrow(k,+)$，其中 $r\ge0$，$1\le j\le n$，且对每个固定群元，除有限多个外其余所有此类特征均消失。记 $S_{n,p}=\begin{cases} \mathbf{N}_{\ge1},&(n,p)=(2,2),\\\\ \mathbf{N}_0,&\text{otherwise}. \end{cases}$ 联合特征映射满射到 $\bigoplus_{s\in S_{n,p}}(k^n,+)$，且每个固定坐标子映射到 $\bigoplus_{s\in S_{n,p}}(k,+)$ 均存在显式群论截面。因此，$\dim_{\mathbb{F}_p} \frac{U_n(k)}{[U_n(k),U_n(k)]\\,U_n(k)^{[p]}} =\aleph_0$。特征 $\chi_{r,j}$ 通过阶为 $N_r=(n-1)(p^{r+1}-1)$ 的射流分解，且对每个 $r\in S_{n,p}$，该阶是最优的。由于 $U_n(k)$ 在 $\operatorname{TA}_n(k)$ 中具有有限指数，有限域上的驯服多项式自同构群仅在一维时有限生成，特别地，这证明了 $n\ge3$ 时的 Maubach–Willems 有限生成猜想。
英文摘要
Let $k=\mathbb F_q$ be a finite field of characteristic $p$, and let \[ U_n(k)=\{F\in\operatorname{TA}_n(k):F(0)=0,\ JF(0)=I_n\}. \] We prove that every special tame polynomial automorphism is repeatedly Cartier-admissible for the de Rham flux class associated with \[ λ=x_1\,dx_2\wedge\cdots\wedge dx_n. \] Lowest-weight projection of the iterated Cartier descents gives additive characters \[ χ_{r,j}:U_n(k)\longrightarrow(k,+), \qquad r\ge0,\quad 1\le j\le n, \] and for every fixed group element all but finitely many of these characters vanish. Put \[ S_{n,p}=\begin{cases} \mathbf N_{\ge1},&(n,p)=(2,2),\\ \mathbf N_0,&\text{otherwise}. \end{cases} \] The joint character map is surjective onto \[ \bigoplus_{s\in S_{n,p}}(k^n,+), \] and every fixed-coordinate submap onto \[ \bigoplus_{s\in S_{n,p}}(k,+) \] admits an explicit group-theoretic section. Consequently, \[ \dim_{\mathbb F_p} \frac{U_n(k)}{[U_n(k),U_n(k)]\,U_n(k)^{[p]}} =\aleph_0. \] The character $χ_{r,j}$ factors through the jet of order \[ N_r=(n-1)(p^{r+1}-1), \] and this order is optimal for every $r\in S_{n,p}$. Since $U_n(k)$ has finite index in $\operatorname{TA}_n(k)$, the tame polynomial automorphism group over a finite field is finitely generated exactly in dimension one. In particular, this proves the Maubach--Willems finite-generation conjecture for $n\ge3$.
