发表机构
Indian Statistical Institute; University of Warwick（印度统计研究所; 华威大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文为Picard群\\(\PSL_2(\ZZ[i])\\)的有限指数子群引入基于高斯Farey镶嵌的Farey符号，证明其可重构子群，并给出基本域计数、Steinberg复形联系及\\(\Gamma_1(2+i)\\)的边界矩阵。
AI 中文摘要
我们为\\(G=\PSL_2(\ZZ[i])\\)的有限指数子群引入Picard Farey符号，其使用由理想八面体构成的\\(\HH^3\\)的高斯Farey镶嵌。一个符号记录有限多个高斯有理八面体出现、它们的局部稳定化子以及有序的余定向面配对。我们证明一个有效的带标记符号能重构有限Picard胞腔作用，从而重构子群，并给出了一系列精确例子。对于指数为\\(n\\)的无挠子群，由完全Farey八面体构成的每个基本域包含\\(n/12\\)个八面体；一个生成树构造给出至多含\\(n/4+1\\)个边配对变换的基本域。我们还把高斯八面体边-面复形等同于\\(\QQ(i)\\)的积分秩二Steinberg表示。因此，相同的装饰几何直接决定系数值Bianchi模符号的有限表示；对于\\(\Gamma_1(2+i)\\)，我们给出完整的\\(3\times4\\)群环边界矩阵。规范约化和有效的Hecke相容约化仍是开放问题。
英文摘要
We introduce Picard Farey symbols for finite-index subgroups of \(G=\PSL_2(\ZZ[i])\), using the Gaussian Farey tessellation of \(\HH^3\) by ideal octahedra. A symbol records finitely many Gaussian-rational octahedral occurrences, their local stabilizers, and ordered cooriented face pairings. We prove that a valid marked symbol reconstructs the finite Picard-cell action and hence the subgroup, and we work out a range of exact examples. For a torsion-free subgroup of index \(n\), every fundamental domain formed from complete Farey octahedra contains \(n/12\) octahedra; a spanning-tree construction gives one with at most \(n/4+1\) side-pairing transformations. We also identify the Gaussian octahedral edge--face complex with the integral rank-two Steinberg presentation for \(\QQ(i)\). Thus the same decorated geometry directly determines a finite presentation of coefficient-valued Bianchi modular symbols; for \(Γ_1(2+i)\) we give the complete \(3\times4\) group-ring boundary matrix. Canonical reduction and an effective Hecke-compatible reduction remain open.
Comments36 pages, 17 figures
