We show that the Connes-Marcolli GL2-system can be represented on the Big Picture, a combinatorial gadget introduced by Conway in order to understand various results about congruence subgroups pictorially. In this representation the time evolution of the GL2-system is implemented by Conway’s distance between projective classes of commensurable lattices. We exploit these results in order to associate quantum statistical mechanical systems to congruence subgroups. This work is motivated by the study of congruence subgroups and their principal moduli in connection with monstrous moonshine.

中文翻译：

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Γ0(N) 等群的非交换几何

我们展示了 Connes-Marcolli GL2 系统可以在大图上表示，这是由康威引入的组合小工具，以便以图形方式理解有关同余子群的各种结果。在这个表示中，GL2 系统的时间演化是通过可公度格的投影类之间的康威距离实现的。我们利用这些结果将量子统计力学系统与同余子群相关联。这项工作的动机是研究同余子群及其与巨大月光有关的主要模量。