modular curve in nLab

Contents

Contents

Idea

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

ℳ ell(ℂ)[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(ℤ)\Gamma(n)\hookrightarrow SL_2(\mathbb{Z}) of the special linear group acting by Möbius transformations.

This has a compactification

ℳ ell(ℂ)[n]↪ℳ ell¯(ℂ)[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

• ℳ ell¯(ℂ)[n] 0≃𝔥/Γ 0(n)\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 ℤ/nℤ\mathbb{Z}/n\mathbb{Z} or order nn;

• ℳ ell¯(ℂ)[n] 1≃𝔥/Γ 1(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 nn-torsion subgroup.

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

• modularity theorem

• The analog of a modular curve with elliptic curves generalized to more general abelian varieties are Shimura varieties.

References

• Wikipedia, Modular curve