geometry, complex numbers, complex line
$dim = 1$: Riemann surface, super Riemann surface
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
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
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.
The analog of a modular curve with elliptic curves generalized to more general abelian varieties are Shimura varieties.
Last revised on July 21, 2015 at 10:51:19. See the history of this page for a list of all contributions to it.