发表机构
Kansas State University（堪萨斯州立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文统一算术与几何方法，分类四元数Hermitian格，证明Hall-Janko构型的唯一性并独立计算其自同构群阶。
AI 中文摘要
我们给出了Hall-Janko单群与二十面体Leech格的统一算术与几何处理。设$\mathcal{I}$为$K=\mathbb{Q}(\sqrt5)$上的标准二十面体极大序。遵循Niemeier格与Leech格的mass公式和Venkov框架，我们对四元数秩为$3$的正定幺模Hermitian $\mathcal{I}$-格进行分类。Hashimoto的mass公式、Siegel-Weil平均和Gundlach的结构定理确定了每个非分裂类的标量theta级数。对Leech最小壳层进行两位置调和分离，证明其三个Hermitian分量是四元数射影$5$-设计。对于范数$2$壳层，完整性及射影矩恢复了完整的角方案及其$525$个正交标架。一个标架的有限码随后饱和了剩余质量，恰好给出分裂类和Tits类，并独立计算出$\operatorname{Aut}_{\mathcal{I}}(L_{\mathrm T})\cong2.J_2$的阶。最后，具有规定的五个射影内积的四元数线系至多有$315$个点，等号恰在Hall-Janko构型（模$P\operatorname{Sp}(3)$）时取得。Hoggar界中的等号提供了设计性质；Cohen-Tits唯一性、黄金角裂变和中心立方矩给出了几何刚性。
英文摘要
We give a unified arithmetic and geometric treatment of the Hall-Janko simple group and the icosian Leech lattice. Let $\mathcal{I}$ be the standard icosian maximal order over $K=\mathbb{Q}(\sqrt5)$. Following the mass-formula and Venkov framework for the Niemeier and Leech lattices, we classify positive definite unimodular Hermitian $\mathcal{I}$-lattices of quaternionic rank $3$. Hashimoto's mass formula, the Siegel-Weil average, and Gundlach's structure theorem determine the scalar theta series of every nonsplit class. A two-place harmonic separation of the Leech minimal shell proves that its three Hermitian components are quaternionic projective $5$-designs. For the norm-$2$ shell, integrality and projective moments recover the complete angle scheme and its $525$ orthogonal frames. The finite code of one frame then saturates the residual mass, giving exactly the split and Tits classes and an independent computation of the order of $\operatorname{Aut}_{\mathcal{I}}(L_{\mathrm T})\cong2.J_2$. Finally, a quaternionic line system with the prescribed five projective inner products has at most $315$ points, with equality precisely for the Hall-Janko configuration up to $P\operatorname{Sp}(3)$. Equality in Hoggar's bound supplies the design property; Cohen-Tits uniqueness, golden-angle fission, and the centered cubic moment give geometric rigidity.
Comments32 pages; LaTeX
