Noncommutative geometry of groups like $\Gamma_0(N)$

TL;DR

This paper connects the Connes-Marcolli $GL_{2}$-system with Conway's Big Picture, enabling a combinatorial and geometric interpretation of the system's dynamics, and explores its relevance to monstrous moonshine.

Contribution

It introduces a novel representation of the $GL_{2}$-system on Conway's Big Picture, linking noncommutative geometry with combinatorial lattice structures for moonshine studies.

Findings

Representation of the $GL_{2}$-system on the Big Picture

Implementation of time evolution via Conway's distance

Application to monstrous moonshine phenomena

Abstract

We show that the Connes-Marcolli $GL_{2}$-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 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 in a way appropriate for the study of monstrous moonshine.