Chamber structure of modular curves $X_{1}(N)$

TL;DR

This paper explores the chamber structure of modular curves $X_{1}(N)$ by linking them to strata of meromorphic differentials, providing formulas for counting chambers and analyzing their combinatorial properties.

Contribution

It introduces a novel geometric-combinatorial approach to compute the genus, punctures, and chamber structure of $X_{1}(N)$ using flat structures of meromorphic differentials.

Findings

Formulas for the number of chambers in the chamber structure

Effective methods for drawing the incidence graph of chambers

A new combinatorial perspective on the topology of modular curves

Abstract

Modular curves $X_{1}(N)$ parametrize elliptic curves with a point of order $N$. They can be identified with connected components of projectivized strata $\mathbb{P}\mathcal{H}(a,-a)$ of meromorphic differentials. As strata of meromorphic differentials, they have a canonical walls-and-chambers structure defined by the topological changes in the flat structure defined by the meromorphic differentials. We provide formulas for the number of chambers and an effective means for drawing the incidence graph of the chamber structure of any modular curve $X_{1}(N)$. This defines a family of graphs with specific combinatorial properties. This approach provides a geometrico-combinatorial computation of the genus and the number of punctures of modular curves $X_{1}(N)$. Although the dimension of a stratum of meromorphic differentials depends only on the genus and the numbers of the singularities, the topological complexity of the stratum crucially depends on the order of the singularities.