复双曲群的分裂域：矩阵提取、厄米特下降与谱域

中文摘要 AI 辅助

设Γ是SU(n,1)的子群，n≥2，在Zariski意义下稠密且包含正则斜驶元A。分裂域问题探究迹与谱能在多大程度上确定Γ的定义域。我们首先证明，迹与A的谱总能在K=Q(trΓ,E(A))上给出线性实现。我们还证明，SU(n,1)中每个具有实迹域的复不可约子群都共轭于SO(n,1)的子群；特别地，该实迹结论无需离散性假设。然而在更高维情形，线性下降本身无法给出标准酉形式：存在厄米特相似障碍，该障碍由Landherr的局部不变量在虚二次域上描述。在SU(3,1)中，我们给出一个Zariski稠密例子，其单谱域为Q(i)，但无法得到标准酉形式，这自然引出了总谱域的概念。我们证明，对于算术群及满足适当局部开性假设的情形，该障碍会消失，且我们给出了偶次局部判据。在四维情形，我们通过显式Schottky群与秩1局部几何分析离散分支，其中完整Bruhat上循环恢复了非迷向Levi全纯性并导出非分裂旋转字。剩余的拟分裂秩2情形无法用该方法处理，因此一般SU(3,1)问题仍未解决。

英文摘要

Let $Γ<\mathrm{SU}(n,1)$, $n\ge2$, be Zariski-dense and contain a regular loxodromic element $A$. The splitting-field problem asks how far traces and spectra determine a field of definition for $Γ$. We show first that traces together with the spectrum of $A$ always give a linear realization over $K=\mathbb{Q}(\textrm{tr}Γ,E(A))$. We also prove that every complex-irreducible subgroup of $\mathrm{SU}(n,1)$ with real trace field is conjugate into $\mathrm{SO}(n,1)$; in particular, no discreteness hypothesis is required for this real-trace conclusion. In higher dimension, however, linear descent does not by itself give the standard unitary form: a Hermitian similarity obstruction remains, described over imaginary quadratic fields by Landherr's local invariants. In $\mathrm{SU}(3,1)$ we give a Zariski-dense example for which the one-spectrum field is $\mathbb{Q}(i)$ but does not yield the standard unitary form. This leads naturally to the total spectral field. We prove that the obstruction disappears for arithmetic groups and under suitable local openness hypotheses, and we give an even-degree local criterion. In dimension four we analyze the discrete branch through explicit Schottky groups and rank-one local geometry. There the full Bruhat cocycle recovers anisotropic Levi holonomy and forces a non-split rotational word. The remaining quasi-split rank-two case is not reached by this method, so the general $\mathrm{SU}(3,1)$ problem remains open.

发表机构

• Instituto de Matemáticas, Universidad Nacional Autónoma de México（墨西哥国立自治大学数学研究所）
• Departamento de Matemática, Instituto de Ciências Exatas, Universidade Federal de Minas Gerais（米纳斯吉拉斯联邦大学理学院数学系）

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