nLab
modular curve

Contents

Idea

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

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

of the upper half plane 𝔥\mathfrak{h} acted on by the n thn^{th} principal congruence subgroup Γ(n)SL 2()\Gamma(n)\hookrightarrow SL_2(\mathbb{Z}) of the special linear group acting by Möbius transformations.

This has a compactification

ell()[n] ell¯()[n] \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

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

References