发表机构
École Normale supérieure de Lyon; Université du Québec à Montréal（里昂高等师范学院; 蒙特利尔魁北克大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究低维紧致平坦流形作为算术双曲流形尖点截面的可能性，利用分类与二次型确定可公度类，并证明多数维度下存在无穷多类，六维有八个例外。
AI 中文摘要
我们研究了维度$3 \leq n \leq 6$的哪些紧致平坦流形可以作为非紧算术双曲$(n+1)$-流形的可公度类中的尖点截面出现。利用低维平坦流形的分类以及其完整表示与有理二次型之间的关系，我们确定了这些维度中每个平坦流形可能的算术可公度类。我们证明，对于$n=3,4,5$，每个平坦$n$-流形都在无穷多个不同的算术可公度类中作为尖点截面出现。在维度六中，除八个例外情况外，同样成立：八个不可定向的平坦$6$-流形仅在唯一的算术可公度类中出现。我们还证明，对于每个$3 \leq n \leq 6$，存在一个算术双曲$(n+1)$-流形的可公度类，其中包含每个平坦$n$-流形作为尖点截面。
英文摘要
We study which compact flat manifolds of dimensions $3 \leq n \leq 6$ occur as cusp cross-sections in commensurability classes of non-compact arithmetic hyperbolic $(n+1)$-manifolds. Using the classification of low-dimensional flat manifolds together with the relationship between their holonomy representations and rational quadratic forms, we determine the possible arithmetic commensurability classes for every flat manifold in these dimensions. We show that every flat $n$-manifold for $n=3,4,5$ occurs as a cusp cross-section in infinitely many distinct arithmetic commensurability classes. In dimension six, the same holds with exactly eight exceptions: eight non-orientable flat $6$-manifolds occur in a unique arithmetic commensurability class. We also show that, for every $3 \leq n \leq 6$, there is a single commensurability class of arithmetic hyperbolic $(n+1)$-manifolds containing every flat $n$-manifold as a cusp cross-section.
