Contents

Ingredients

Concepts

Constructions

Examples

Theorems

# Contents

## Idea

Over the complex numbers an elliptic curve (hence a complex torus) may be, and often is, presented as a quotient of the complex plane by a framed lattice, determined by a point $\tau$ in the upper half plane $\mathfrak{h}$. But indeed this data determines a framed elliptic curve, namely the underlying curve $\Sigma$ together with the “edges along which it is glued” by this construction. These are equivalently a choice of ordered basis

$(a_1,a_2)\in H_1(\Sigma,\mathbb{Z})$

of the first ordinary homology of $\Sigma$ with integer coefficients (and vanishing intersection number). The special linear group $SL_2(\mathbb{Z})$ naturally acts on this data (by Möbius transformations on $\tau$) and the quotient projection exhibits an infinite covering (atlas) of the moduli stack of elliptic curves over the complex numbers

$\mathfrak{h} \longrightarrow \mathfrak{h}//SL_2(\mathbb{C}) \simeq \mathcal{M}_{ell}(\mathbb{C}) \,.$

The concept of level structure on an elliptic curve is a structure weaker than that of a framing which analogously gives a finite covering. Instead of considering cycles in integral homology, a level $n$-structure for natural number $B$ is given by cycles in homology with coefficients just in the cyclic group $\mathbb{Z}/n\mathbb{Z}$ (e.g. Hain 08, def. 4.6).

On such level-$n$ data now acts instead just the group $SL_2(\mathbb{Z}/n\mathbb{Z})$. The kernel of the projection maps is called the level $n$-subgroup (an example of a congruence subgroup)

$\Gamma(n) \to SL_2(\mathbb{Z}) \to SL_2(\mathbb{Z}/n\mathbb{Z}) \,.$

There is then a moduli space of complex elliptic curves equipped with level $n$-structure

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

called the modular curve, and this is now a finite cover (of rank the order of the finite group $SL_2(\mathbb{Z}/n\mathbb{Z})$) of the actual moduli stack of complex elliptic curves

$\mathcal{M}_{ell}[n](\mathbb{C}) \longrightarrow \mathcal{M}_{ell}[n](\mathbb{C})//SL_2(\mathbb{Z}/n\mathbb{Z}) \simeq \mathcal{M}_{ell}(\mathbb{C}) \,.$

## Definition

### Congruence subgroups

Let $n \in \mathbb{N}$ be a natural number. Write

$p_n \;\colon\; SL_2(\mathbb{Z}) \to SL_2(\mathbb{Z}/\mathbb{Z}_n)$

for the projection from the special linear group induced by the quotient projection $\mathbb{Z} \to \mathbb{Z}/n\mathbb{Z}$ to the cyclic group.

The congruence subgroups of the special linear group $SL_2(\mathbb{Z})$ (essentially the modular group) are defined as follows.

The principal congruence subgroup is

$\Gamma(n) \coloneqq ker(p_n) = p_n^{-1}\left(\left\{\array{ 1 & 0 \\ 0 & 1}\right\}\right)$

The other two are

$\Gamma_0(n) \coloneqq p_n^{-1}\left(\left\{\array{ \ast & \ast \\ 0 & \ast}\right\}\right)$
$\Gamma_1(n) \coloneqq p_n^{-1}\left(\left\{\array{ 1 & \ast \\ 0 & \ast}\right\}\right)$

## Properties

### Relation to spin structures on elliptic curves

For elliptic curves over the complex numbers (complex oriented pointed tori) the congruence subgroup $\Gamma_0(2)$ has the interpretation as being precisely the subgroup of the modular group which preserves one of the “NS-R” spin structures.

In detail, the elliptic curve $\Sigma$, being framed has a canonical spin structure given by the trivial double cover. The space of all spin structures is a torsor over $H^1(\Sigma, \mathbb{Z}/2\mathbb{Z}) \simeq [\pi_1(\sigma), \mathbb{Z}/2\mathbb{Z}] \simeq [\mathbb{Z} \times \mathbb{Z}, \mathbb{Z}/2\mathbb{Z}] \simeq (\mathbb{Z}/2\mathbb{Z})^2$. In terms of this action the canonical one is labeled $(0,0)$ and then there are three more, labeled $(1,0)$, $(0,1)$ and $(1,1)$. The modular group acts on these via the quotient map $p_2 \;\colon\; SL_2(\mathbb{Z}) \to SL_{2}(\mathbb{Z}/2\mathbb{Z})$. Hence it preserves $(0,0)$ and mixes the other three spin structures. Precisely $\Gamma_0(2)$ preserves $(1,0)$ (and an isomorphic subgroup of course preserves $(0,1)$). The principal congruence subgroup $\Gamma(2)$ is the one which preserves all four spin structures jointly.

In terms of type II string theory the spin structure $(1,0)$ is called the “NS-R boundary condition” for the spinors. The partition function of the type II superstring “in the NS-R sector” is therefore (at best, indeed it is, being the universal Ochanine elliptic genus) a modular form not for the full modular group, but for $\Gamma_0(2)$ (Witten 87a, below (13)). For more on this see at Witten genus – Modularity – For the type II string. The homotopy-theoretic refinement of this involves tmf0(2), see at spin orientation of Ochanine elliptic cohomology.

## Properties

### Topological modular forms with level structure

The construction of topological modular forms (tmf) may be generalized to curves with level structure (Mahowald-Rezk 09). A systematic kind of “modular equivariant elliptic cohomology” in this sense is discussed in (Hill-Lawson 13).

### Over the complex numbers

The principal congruence subgroups are discussed for instance in

The relation of $\Gamma_1(2)$ to spin structures is discussed for instance in

• Daniel Freed, pages 24-25 of On determinant line bundles, 1987 (pdf)

• Edward Witten, Elliptic Genera And Quantum Field Theory , Commun.Math.Phys. 109 525 (1987) (Euclid)

### Over general base rings

The concept of level structure on an elliptic curve is due to

• Nicholas M. Katz, Barry Mazur, Arithmetic moduli of elliptic curves, Princeton University Press, Princeton, NJ, 1985

Review of the definition includes

General discussion is in

Discussion of the corresponding moduli stack and its tmf$(n)$-spectrum is in

Specifically Level-2 structure in this context is discussed in