complex geometry

# Contents

## Idea

A modular curve is a moduli space of elliptic curves over the complex numbers equipped with level-n structure, for some $n \in \mathbb{N}$. Concretely this is equivalent to the quotient

$\mathcal{M}_{ell}(\mathbb{C})[n] \coloneqq \mathfrak{h}/\Gamma(n)$

of the upper half plane $\mathfrak{h}$ acted on by the $n^{th}$ principal congruence subgroup $\Gamma(n)\hookrightarrow SL_2(\mathbb{Z})$ of the special linear group acting by Möbius transformations.

This has a compactification

$\mathcal{M}_{ell}(\mathbb{C})[n] \hookrightarrow \mathcal{M}_{\overline{ell}}(\mathbb{C})[n]$

and often that is referred to by default as the modular curve.

The quotients by the other two congruence subgroups are

• $\mathcal{M}_{\overline{ell}}(\mathbb{C})[n]_0 \simeq \mathfrak{h}/\Gamma_0(n)$ – the moduli space of complex elliptic curves equipped with a cyclic subgroup $\mathbb{Z}/n\mathbb{Z}$ or order $n$;

• $\mathcal{M}_{\overline{ell}}(\mathbb{C})[n]_1 \simeq \mathfrak{h}/\Gamma_1(n)$ – the moduli space of complex elliptic curves equipped with an element (a point) in an $n$-torsion subgroup.

By construction, these modular curves provide covers (atlases) for the moduli stack of elliptic curves $\mathcal{M}_{ell}(\mathbb{C})$ over the complex numbers.