# Contents

## Idea

For the moment see A Survey of Elliptic Cohomology - elliptic curves for more.

An elliptic curve $E \to Spec \mathbb{C}$ is determined, up to non-canonical isomorphism, by its j-invariant

$j(E) \in \mathbb{C} \,.$

Here every complex number appears as a value, and therefore the moduli space of elliptic cuves a priori is not compact.

A compactification of the moduli space is obtained by including also elliptic curves with nodal singularity?.

## Description over the complex numbers

### Upper half plane

The upper half plane $\mathfrak{h}$ is in bijection with framed lattices in the complex plane $\mathbb{C}$, which in turn is in bijection with isomorphism classes of framed elliptic curves over $\mathbb{C}$

$\mathfrak{h} \simeq \{framed\;lattices\;in\;\mathbb{C}\} \simeq \{framed\;elliptic\;curves\;over\;\mathbb{C}\}/_\sim$

and we have

$\{elliptic\;curves\;over\;\mathbb{C}\}_\sim \simeq \mathfrak{h}/{SL_2(\mathbb{Z})}$

where the special linear group over the integers

$SL_2(\mathbb{Z}) = \left\{ \left(\array{a & b \\ c & d }\right)| a d - c d = 1\right\}$

acts by

$\tau \mapsto \frac{a \tau + b}{c \tau + d} \,.$

### The naive moduli space and its problems

###### Definition

Write

$M_{1,1} := \mathfrak{h}/SL_2(\mathbb{Z})$

for the plain quotient of the upper half plane by the above group action.

###### Definition

A homolorphic family of elliptic curves over a complex manifold $T$ is

• a holomorphic function $\pi : X \to T$

• together with a section $s : T \to X$ of $\pi$ such that for any $t \in T$ the pair $(X_t, s(t))$ is an elliptic curve (using the first definition above).

For every family

$\array{ X \\ \downarrow^{\mathrlap{\pi}} \\ T }$

we would like to have $F \to M_{1,1}$ such that there is a pullback

$\array{ X \simeq \phi^* F &\longrightarrow& F \\ \downarrow & & \downarrow \\ T &\stackrel{\phi}{\longrightarrow}& M_{1,1} }$

where

$\phi: t \mapsto [X_t, s(t)]$

such that

• $\phi : T \to M_{1,1}$ is a holomorphic map

• every holomorphic map $T \to M_{1,1}$ corresponds to a family over $t$;

• there is a universal family over $M_{1,1}$

This is impossible . One can construct explicit counterexamples. These counterexamples involve elliptic curves with nontrivial automorphisms.

For instance

$\{ (x,y,z) \in \mathbb{P}^2 \times X : y^2 = x(x-1)(x-\lambda) \} \to X := \mathbb{P}^1 - \{0,1,\infty\}$

but see the discussion at moduli space for a discussion of the statement “it’s te automorphisms that prevent the moduli space from existing”

### Moduli space of framed elliptic curves

consider

$\mathbb{Z}^2 \hookrightarrow \mathbb{C} \times \mathfrak{h}$

given by

$(m,n) : (z,\tau) \mapsto (z + m \tau + n, \tau)$

Then consider the family

$\array{ E := \mathbb{C}/_{\mathbb{Z}^2} \times \mathfrak{h} \\ \downarrow \\ \mathfrak{h} }$

is a family of elliptic curves over $\mathfrak{h}$

and $E_\tau = \mathbb{C}/{\Lambda_\tau}$ with

$\Lambda_{\tau} := \mathbb{Z}\cdot 1 \oplus \mathbb{Z}\cdot \tau$

is a family of framed elliptic curves.

###### Proposition

The space $\mathfrak{h}$ with the family $E \to \mathfrak{h}$ is a fine moduli space for framed elliptic curves?.

Consider any map $\phi : T \to \mathfrak{h}$

with pullback of the universal family

$\array{ X \stackrel{?}{\to} \phi^* E &\to & E \\ \downarrow && \downarrow \\ T &\stackrel{\phi}{\to}& \mathfrak{h} }$

claim for every point $t \in T$ there is an open neighbourhood $t_0 \in U \hookrightarrow T$ such that one can choose 1-forms $\omega_t$ on $X_\tau$ which vary holomorphically with respect to $t$.

Notice that locally every family of elliptic curves is framed (since we can locally extend a choice of basis for $H_1$). So

$\array{ && \mathfrak{h} \\ && \downarrow^{SL_2(\mathbb{Z})} \\ M_{1,1} &\stackrel{Id}{\to}& M_{1,1} }$

at $i$ and $\rho = e^{2\pi i/6}$ , $C = \{\pm I\}$

isn’t locally liftable at $i$ and $\rho$ so it is not a univresal family of unframed curves.

### Moduli stack/orbifold of elliptic curves

###### Definition

Consider the global quotent stack? orbifold

$\mathcal{M}_{1,1} := \mathfrak{h}//SL_2(\mathbb{Z})$

of the upper half plane by the action of the special linear group over the integers.

This is the moduli stack of elliptic curves.

### Compactified moduli stack

Consider the complex analytic parameteritation over the annulus

$\{q \in \mathbb{C} | 0 \lt {\vert q \vert} \lt 1 \}$

of elliptic curves

$E_q \coloneqq \mathbb{C}/q^{\mathbb{Z}} \,.$

This has an extension to the origin, where $E_0$ is a nodal curve. Algebraically, in a formal neighbourhood? of the origin, hence over $Spec \mathbb{Z}[ [q] ]$, this is the Tate curve.

e.g. (Lurie, section 4.3).

## Properties

### Cohomology

###### Proposition
$H_1(\mathcal{M}_{1,1}, \mathbb{Z}) = \mathbb{Z}/12\mathbb{Z}$
$H^1(\mathcal{M}_{1,1}, \mathbb{Z}) = 0$
$H^2(\mathcal{M}_{1,1}, \mathbb{Z}) = \mathbb{Z}/12 \mathbb{Z}$
$H_\bullet(\mathcal{M}_{1,1}, \mathbb{Q}) \simeq H_\bullet(M_{1,1}, \mathbb{Q})$

and similarly for integral cohomology

$\chi(\mathcal{M}_{1,1}) = -\frac{1}{12}$
$Pic(\mathcal{M}_{1,1}) \simeq \mathbb{Z}/12\mathbb{Z}$

## References

An introduction is for instance in

• Andre Henriques, The moduli stack of elliptic curves (pdf) in Topological modular forms Talbot workshop 2007 (web)

Revised on November 12, 2013 11:58:05 by Urs Schreiber (188.200.54.65)