左平移相似轨道有限样本刚性及非对角二次实现

中文摘要 AI 辅助

我们研究交换环上相似轨道左平移的有限样本刚性。设R为满足2∈R^×的交换环，C∈Mat_2(R)为循环矩阵。给定矩阵G_i=FK_i，其中F∈GL_2(R)为公共矩阵，底层点K_i位于Orb(C)中，我们确定有限平移样本何时具有与完整平移轨道完全相同的乘子歧义。关键结构结果是轨道差张成sl_2(R)，这给出了左稳定子的标量描述。随后我们给出一个行列式饱和准则，直接以矩阵G_i表达，在该准则下有限样本一致性归结为迹方程和一个行列式条件。两个自由秩二模对上的非对角二次算子精确实现了平移轨道模型。在域上，我们在饱和准则下获得尖锐样本阈值；在有限域上，我们推导出正则半单轨道和非零幂零轨道的精确饱和概率。椭圆同源性的挠限制为秩二模框架提供了自然的算术实现。

英文摘要

We study finite-sample rigidity for left translates of similarity orbits over commutative rings. Let \(R\) be a commutative ring with \(2\in R^\times\), and let \(C\in\operatorname{Mat}_2(R)\) be cyclic. Given matrices \(G_i=FK_i\), where \(F\in\operatorname{GL}_2(R)\) is common and the underlying points \(K_i\) lie in \(\operatorname{Orb}(C)\), we determine when a finite translated sample has exactly the same multiplier ambiguity as the complete translated orbit. The key structural result is that the orbit differences span \(\mathfrak{sl}_2(R)\), which yields a scalar description of the left stabilizer. We then give a determinantal saturation criterion, expressed directly in the matrices \(G_i\), under which finite-sample consistency reduces to trace equations and one determinant condition. Off-diagonal quadratic operators on pairs of free rank-two modules realize the translated-orbit model exactly. Over fields we obtain sharp sample thresholds under the saturation criterion, and over finite fields we derive exact saturation probabilities for regular semisimple and nonzero nilpotent orbits. Torsion restrictions of elliptic isogenies provide a natural arithmetic realization of the rank-two module framework.

发表机构

• National Institute of Technology, Tokyo College（东京工业高等专门学校）
• Institute of Science Tokyo（东京科学大学）

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