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modular curve

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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

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

  • ell¯()[n] 1𝔥/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 ell()\mathcal{M}_{ell}(\mathbb{C}) over the complex numbers.

References

Last revised on July 21, 2015 at 06:51:19. See the history of this page for a list of all contributions to it.