Contents

Idea

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

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

$j\left(E\right)\in ℂ\phantom{\rule{thinmathspace}{0ex}}.$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 $𝔥$ is in bijection with framed lattices in the complex plane $ℂ$, which in turn is in bijection with isomorphism classes of framed elliptic curves over $ℂ$

$𝔥\simeq \left\{\mathrm{framed}\phantom{\rule{thickmathspace}{0ex}}\mathrm{lattices}\phantom{\rule{thickmathspace}{0ex}}\mathrm{in}\phantom{\rule{thickmathspace}{0ex}}ℂ\right\}\simeq \left\{\mathrm{framed}\phantom{\rule{thickmathspace}{0ex}}\mathrm{elliptic}\phantom{\rule{thickmathspace}{0ex}}\mathrm{curves}\phantom{\rule{thickmathspace}{0ex}}\mathrm{over}\phantom{\rule{thickmathspace}{0ex}}ℂ\right\}{/}_{\sim }$\mathfrak{h} \simeq \{framed\;lattices\;in\;\mathbb{C}\} \simeq \{framed\;elliptic\;curves\;over\;\mathbb{C}\}/_\sim

and we have

$\left\{\mathrm{elliptic}\phantom{\rule{thickmathspace}{0ex}}\mathrm{curves}\phantom{\rule{thickmathspace}{0ex}}\mathrm{over}\phantom{\rule{thickmathspace}{0ex}}ℂ{\right\}}_{\sim }\simeq 𝔥/{\mathrm{SL}}_{2}\left(ℤ\right)$\{elliptic\;curves\;over\;\mathbb{C}\}_\sim \simeq \mathfrak{h}/{SL_2(\mathbb{Z})}

where the special linear group over the integers

${\mathrm{SL}}_{2}\left(ℤ\right)=\left\{\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\mid ad-cd=1\right\}$SL_2(\mathbb{Z}) = \left\{ \left(\array{a & b \\ c & d }\right)| a d - c d = 1\right\}

acts by

$\tau ↦\frac{a\tau +b}{c\tau +d}\phantom{\rule{thinmathspace}{0ex}}.$\tau \mapsto \frac{a \tau + b}{c \tau + d} \,.

The naive moduli space and its problems

Definition

Write

${M}_{1,1}:=𝔥/{\mathrm{SL}}_{2}\left(ℤ\right)$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 $\left({X}_{t},s\left(t\right)\right)$ is an elliptic curve (using the first definition above).

For every family

$\begin{array}{c}X\\ {↓}^{\pi }\\ T\end{array}$\array{ X \\ \downarrow^{\mathrlap{\pi}} \\ T }

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

$\begin{array}{ccc}X\simeq {\varphi }^{*}F& ⟶& F\\ ↓& & ↓\\ T& \stackrel{\varphi }{⟶}& {M}_{1,1}\end{array}$\array{ X \simeq \phi^* F &\longrightarrow& F \\ \downarrow & & \downarrow \\ T &\stackrel{\phi}{\longrightarrow}& M_{1,1} }

where

$\varphi :t↦\left[{X}_{t},s\left(t\right)\right]$\phi: t \mapsto [X_t, s(t)]

such that

• $\varphi :T\to {M}_{1,1}$ 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

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

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

given by

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

Then consider the family

$\begin{array}{c}E:=ℂ{/}_{{ℤ}^{2}}×𝔥\\ ↓\\ 𝔥\end{array}$\array{ E := \mathbb{C}/_{\mathbb{Z}^2} \times \mathfrak{h} \\ \downarrow \\ \mathfrak{h} }

is a family of elliptic curves over $𝔥$

and ${E}_{\tau }=ℂ/{\Lambda }_{\tau }$ with

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

is a family of framed elliptic curves.

Proposition

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

Consider any map $\varphi :T\to 𝔥$

with pullback of the universal family

$\begin{array}{ccc}X\stackrel{?}{\to }{\varphi }^{*}E& \to & E\\ ↓& & ↓\\ T& \stackrel{\varphi }{\to }& 𝔥\end{array}$\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 ${t}_{0}\in U↪T$ 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

$\begin{array}{ccc}& & 𝔥\\ & & {↓}^{{\mathrm{SL}}_{2}\left(ℤ\right)}\\ {M}_{1,1}& \stackrel{\mathrm{Id}}{\to }& {M}_{1,1}\end{array}$\array{ && \mathfrak{h} \\ && \downarrow^{SL_2(\mathbb{Z})} \\ M_{1,1} &\stackrel{Id}{\to}& M_{1,1} }

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

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

${ℳ}_{1,1}:=𝔥//{\mathrm{SL}}_{2}\left(ℤ\right)$\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

$\left\{q\in ℂ\mid 0<\mid q\mid <1\right\}$\{q \in \mathbb{C} | 0 \lt {\vert q \vert} \lt 1 \}

of elliptic curves

${E}_{q}≔ℂ/{q}^{ℤ}\phantom{\rule{thinmathspace}{0ex}}.$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 $\mathrm{Spec}ℤ\left[\left[q\right]\right]$, this is the Tate curve.

e.g. (Lurie, section 4.3).

Properties

Cohomology

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

and similarly for integral cohomology

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