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A Survey of Elliptic Cohomology - elliptic curves

This is a sub-entry of

• A Survey of Elliptic Cohomology

and

• Gromov-Witten invariants

see there for background and context.

This entry contains a basic introduction to elliptic curves and their moduli spaces.

Previous:

• A Survey of Elliptic Cohomology - cohomology theories

• A Survey of Elliptic Cohomology - formal groups and cohomology

• A Survey of Elliptic Cohomology - E-infinity rings and derived schemes

Next:

• A Survey of Elliptic Cohomology - equivariant cohomology

the following are rough unpolished notes taken more or less verbatim from some seminar talk – needs attention, meaning: somebody should go through this and polish

---

elliptic curves

Definition An elliptic curve over $ℂ$ is equivalently

• a Riemann surface $X$ of genus 1 with a fixed point $P\in X$

• a quotient $ℂ/\Lambda$ where $\Lambda$ is a lattice in $ℂ$;

• a compact complex Lie group of dimension 1.

• a smooth algebraic curve of degree 3 in $𝒫$.

Remark The third definition is the one that is easiest to generalize. For our simple purposes, though, the second one will be the most convenient.

From the second definition it follows that to study the moduli space of elliptic curves it suffices to study the moduli space of lattices in $ℂ$.

Definition A framed elliptic curve? is an elliptic curve $(X,P)$ (in the sense of the first definition above) together with an ordered basis $(a,b)$ of $H_1(X,\mathbb{Z})$ with $(a\cdot b)=1$

A framed lattice in $ℂ$ is a lattice $\Lambda$ together with an ordered basis $({\lambda }_1,{\lambda }_2)$ of $\Lambda$ such that $\mathrm{Im}({\lambda }_2/{\lambda }_1)>0$.

(So turning ${\lambda }_1$ to ${\lambda }_2$ in the plane means going counterclockwise).

moduli spaces of elliptic curves

this implies that

the upper half plane? $𝔥$ is in bijection with framed lattices in $ℂ$ which in turn is in bijection with isomorphism classes of framed elliptic curves over $ℂ$

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

and we have

$$\{\mathrm{elliptic}\mathrm{curves}\mathrm{over}ℂ{\}}_{\sim }\simeq 𝔥/{\mathrm{SL}}_2(\mathbb{Z})$$\{elliptic curves over \mathbb{C}\}_\sim \simeq \mathfrak{h}/{SL_2(\mathbb{Z})}

where ${\mathrm{SL}}_2(\mathbb{Z})=\{\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\mid ad-cd=1\}$ acts by

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

Claim the quotient $𝔥{/}_{{\mathrm{SL}}_2(\mathbb{Z})}$ is biholomorphic? to the disk and has a unique structure of a Riemann surface which makes the quotient map $𝔥\to 𝔥/{\mathrm{SL}}_2(\mathbb{Z})$ a holomorphic map

warning possibly something wrong here, audience doesn’t believe the bit about the disk

definition write $M_{1,1}:=𝔥/{\mathrm{SL}}_2(\mathbb{Z})$

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

• a holomorphic map $\pi :X\to T$

• 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

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

we would like to have $F\to M_{1,1}$

$$\begin{array}{ccc}X\simeq {\varphi }^{*}F& \to & F\\ \downarrow & & \downarrow \\ T& \to & M_{1,1}\end{array}$$\array{ X \simeq \phi^* F &\to& F \\ \downarrow & & \downarrow \\ T &\to& M_{1,1} }

where

$$\varphi :t\mapsto [X_t,s(t)]$$\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 involved elliptic curves with nontrivial automorphisms.

For instance

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

consider

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

given by

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

Then consider the family

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

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

is a family of framed elliptic curves.

fact the space $𝔥$ with the family $E\to 𝔥$ is a fine moduli space for framed elliptic curve?s.

Consider any map $\varphi :T\to 𝔥$

with pullback of the universal family

$$\begin{array}{ccc}X\stackrel{?}{\to }{\varphi }^{*}E& \to & E\\ \downarrow & & \downarrow \\ T& \stackrel{\varphi }{\to }& 𝔥\end{array}$$\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 $t_0\in U\hookrightarrow T$ 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

$$\begin{array}{ccc}& & 𝔥\\ & & {\downarrow }^{{\mathrm{SL}}_2(\mathbb{Z})}\\ M_{1,1}& \stackrel{\mathrm{Id}}{\to }& M_{1,1}\end{array}$$\array{ && \mathfrak{h} \\ && \downarrow^{SL_2(\mathbb{Z})} \\ M_{1,1} &\stackrel{Id}{\to}& M_{1,1} }

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

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

orbifolds

definition A basic pointed orbifold (basic meaning global) is a triple $X//\Gamma :=(X,\Gamma ,\rho )$, where

• $X$ is a connected and simply connected topological space (or in other variants a complex manifold or whatever is under consideration)

• $\Gamma$ is a discrete group

• $\rho :\Gamma \to \mathrm{Aut}(X)$ is a group homomorphism

(here “pointed” because we specified the action $\rho$ instead of its iso-class under the following morphisms)

A morphism from $(X,\gamma ,\rho )$ to $(X\prime ,\Gamma \prime ,\rho \prime )$ is a pair

$$(f,\varphi )$$(f,\phi)

where

• $f:X\to X\prime$ is a continuous map

• $\varphi :\Gamma \to \Gamma \prime$ is a group homomorphism

such that for all $\gamma \in \Gamma$

$$\begin{array}{ccc}X& \stackrel{f}{\to }& X\prime \\ {\downarrow }^{\gamma }& & {\downarrow }^{\varphi (\gamma )}\\ X& \stackrel{f}{\to }& X\prime \end{array}$$\array{ X &\stackrel{f}{\to}& X' \\ \downarrow^{\gamma} && \downarrow^{\phi(\gamma)} \\ X &\stackrel{f}{\to}& X' }

This really leads an enlargement of the plain category of spaces:

remark We have a faithful embedding of spaxces into orbifolds defined this way: for any connected semi-locally simply connected space $X$ with universal cover $\tilde{X}$ we have

$$X\mapsto \tilde{X}//{\pi }_1(X)$$X \mapsto \tilde X //\pi_1(X)

warning notice all the simply-connectedness assumoptions above for making sense of this

remark let $X$ be a nice topological space. Let $G={\pi }_1(X)$ be its first homotopy group and let a discrete group $\Gamma$ action on $X$.

then define

$$\tilde{\Gamma }:=\{(\gamma ,g)\in \Gamma \times \mathrm{Aut}(\tilde{X})\mid \begin{array}{ccc}\tilde{X}& \stackrel{g}{\to }& \tilde{X}\\ {\downarrow }^p& & {\downarrow }^p\\ X& \stackrel{\gamma }{\to }& X\end{array}\}$$\tilde \Gamma := \left\{ (\gamma,g) \in \Gamma \times Aut(\tilde X) | \array{ \tilde X &\stackrel{g}{\to}& \tilde X \\ \downarrow^p && \downarrow^p \\ X &\stackrel{\gamma}{\to}& X } \right\}

then we have an exact sequence

$$1\to G\to \tilde{\Gamma }\to \Gamma \to 1$$1 \to G \to \tilde \Gamma \to \Gamma \to 1

where $G\to \tilde{\Gamma }$ is given by $(g\mapsto (1,g))$ and $\tilde{\Gamma }\to \Gamma$ by $(\gamma ,g)\mapsto \gamma$.

definition

For an orbifold $(X,\Gamma ,\rho )$ write $I\times (X,\Gamma ,\rho ):=(I\times X,\Gamma ,\rho )$.

Then a homotopy from $(f,\varphi )$ to $(f\prime ,\varphi \prime ):(X,\Gamma ,\rho )\to (X\prime ,\Gamma \prime ,\rho \prime )$

is a map

$$(F,\Psi ):I\times (X,\Gamma ,\rho )\to (X\prime ,\Gamma \prime ,\rho \prime )$$(F,\Psi) : I \times (X, \Gamma, \rho) \to (X', \Gamma', \rho')

such that

• $\Psi =\varphi =\varphi \prime$

• $f(-)=F(0,-)$, $f\prime (-)=F(1,-)$

now write

$$S^1:=(ℝ,\mathbb{Z})$$S^1 := (\mathbb{R}, \mathbb{Z})

(the circle regarded as a global orbifold)

definition

The first homotopy group for our definition of orbifold is:

$${\pi }_1(X//\gamma )\simeq \{\mathrm{homotopy}\mathrm{classes}\mathrm{of}\mathrm{maps}{S}^1\to X//\Gamma \}$$\pi_1(X//\gamma) \simeq \{ homotopy classes of maps S^1 \to X//\Gamma \}

exercise show that this is $\cdots \simeq \Gamma$

(recall again the simply-connectness assumoption!!)

definition* A morphism

$$(f,\varphi ):(X,\Gamma )\to (X\prime ,\Gamma \prime )$$(f,\phi) : (X,\Gamma) \to (X',\Gamma')

is a weak homotopy equivalence if $\varphi$ is an ismorphism and $H_{•}(f):H_{•}(X)\to X_{•}(X\prime )$.

note Let $E\Gamma$ be a contractible space on which $\Gamma$ acts properly, dic. and free, then

$$(E\Gamma \times X,\Gamma )\stackrel{(f,\varphi )}{\to }(X,\Gamma )$$(E \Gamma \times X, \Gamma) \stackrel{(f,\phi)}{\to} (X,\Gamma)

with $\varphi ={\mathrm{Id}}_{\Gamma }$ and $f$ the projection is a weak homotopy equivalence.

definition a local system $V$ on $(X,\Gamma )$ with fiber $V$ a group homomorphism $\Gamma \to \mathrm{Aut}(V)$ with

definition

Introduce the following notation for homotopy groups, homology and integral cohomology of our orbifolds with coefficients in a local system:

• ${\pi }_n(X//\Gamma )$ := \pi_n(X)$\mathrm{for}$n \geq 2$

• $H_{•}(X//\Gamma ,V):=H_{•}(E\Gamma {\times }_{Gama}X,V)$

• $H^{•}(X//\Gamma ,V):=H^{•}(E\Gamma {\times }_{Gama}X,V)$

example $*//\Gamma$ has a weak homotopy equivalence to the classifying space $ℬ\Gamma$

it follows that for local system $V$ we have

$$H^{•}(*//\Gamma ,V)=H_{\mathrm{gp}}^{•}(\Gamma ,V)$$H^\bullet({*}//\Gamma, V) = H^\bullet_{gp}(\Gamma,V)

where on the right we have group cohomology

---

We have all kinds of constructions on orbifolds by saying they are structures on $X$ with suitable extension of the action of $\Gamma$ to them

A vector bundle on an orbifold $V\to X//\Gamma$ is a vector bundle $V\to X$ with isomorphism action by $\Gamma$ specified, covering that on $X$.

for instance the tangent bundle of $X//\Gamma$ is given by $(TX)//\Gamma \to X//\Gamma$ in the obvious way.

definition say that $\Gamma$ acts virtually freely if $\exists$ a finite index subgroup $\Gamma \prime$ of $\Gamma$ which acts freely on $X$.

note ${\mathrm{SL}}_2(\mathbb{Z})$ acts virtually freely on $𝔥$

$${\mathrm{SL}}_2(\mathbb{Z})[m]\to {\mathrm{SL}}_2(\mathbb{Z})\to {\mathrm{SL}}_2(\mathbb{Z}/m\mathbb{Z})$$SL_2(\mathbb{Z})[m] \to SL_2(\mathbb{Z}) \to SL_2(\mathbb{Z}/ m \mathbb{Z})

Let $\Gamma \prime <\Gamma$ be a finite index subgroup which acts freely on $X$.

set

$$X\prime :=X//\Gamma \prime ;$$X' := X// \Gamma';

the map

$$X/\Gamma \prime \to X//\Gamma$$X/\Gamma' \to X//\Gamma

must be viewed as an unramified covering of degree $[\Gamma :\Gamma \prime ]$.

supposedly important statement

definition

the Euler characteristic of a global orbifold is

$$\chi (X//\Gamma ):=\frac{1}{[\Gamma :\Gamma \prime ]}\chi (X/\Gamma \prime )$$\chi(X//\Gamma) := \frac{1}{[\Gamma: \Gamma']} \chi(X/\Gamma')

compare groupoid cardinality

moduli stack/orbifold of elliptic curves

definition

Define now the global orbifold

$${ℳ}_{1,1}:=𝔥//{\mathrm{SL}}_2(\mathbb{Z})$$\mathcal{M}_{1,1} := \mathfrak{h}//SL_2(\mathbb{Z})

proposition

$$H_1({ℳ}_{1,1},\mathbb{Z})=\mathbb{Z}/12\mathbb{Z}$$H_1(\mathcal{M}_{1,1}, \mathbb{Z}) = \mathbb{Z}/12\mathbb{Z}

$$H^1({ℳ}_{1,1},\mathbb{Z})=0$$H^1(\mathcal{M}_{1,1}, \mathbb{Z}) = 0

$$H^2({ℳ}_{1,1},\mathbb{Z})=\mathbb{Z}/12\mathbb{Z}$$H^2(\mathcal{M}_{1,1}, \mathbb{Z}) = \mathbb{Z}/12 \mathbb{Z}

$$H_{•}({ℳ}_{1,1},\mathbb{Q})\simeq H_{•}(M_{1,1},\mathbb{Q})$$H_\bullet(\mathcal{M}_{1,1}, \mathbb{Q}) \simeq H_\bullet(M_{1,1}, \mathbb{Q})

and similarly for integral cohomology

$$\chi ({ℳ}_{1,1})=-\frac{1}{12}$$\chi(\mathcal{M}_{1,1}) = -\frac{1}{12}

$$\mathrm{Pic}({ℳ}_{1,1})\simeq \mathbb{Z}/12\mathbb{Z}$$Pic(\mathcal{M}_{1,1}) \simeq \mathbb{Z}/12\mathbb{Z}

topological invariants of the moduli stack

Since the upper half plane is contractible, the homotopy type of $𝔥//{\mathbb{Z}}_2$ are the same as that of $*//{\mathbb{Z}}_2$ and similarly for the (group)cohomology

$$H^{•}({ℳ}_{1,1},\mathbb{Z})=H•({\mathrm{SL}}_2(\mathbb{Z}),\mathbb{Z})$$H^\bullet(\mathcal{M}_{1,1}, \mathbb{Z}) = H\bullet(SL_2(\mathbb{Z}), \mathbb{Z})

and similalry for homology.

In particular

$$H_1({ℳ}_{1,1},\mathbb{Z})\simeq {\mathrm{SL}}_2(\mathbb{Z}{)}^{\mathrm{ab}}\simeq \mathbb{Z}/12\mathbb{Z}$$H_1(\mathcal{M}_{1,1}, \mathbb{Z}) \simeq SL_2(\mathbb{Z})^{ab} \simeq \mathbb{Z}/12\mathbb{Z}

for all $m\in \mathbb{N}$ then

$$1\to {\mathrm{SL}}_2(\mathbb{Z})[m]\to {\mathrm{SL}}_2(\mathbb{Z})\to {\mathrm{SL}}_2(\mathbb{Z}/m\mathbb{Z})\to 1$$1 \to SL_2(\mathbb{Z})[m] \to SL_2(\mathbb{Z}) \to SL_2(\mathbb{Z}/m \mathbb{Z}) \to 1

so that

$$H^1({ℳ}_{1,1},\mathbb{Z})=0$$H^1(\mathcal{M}_{1,1}, \mathbb{Z}) = 0

fact the group ${\mathrm{SL}}_2(\mathbb{Z})[m]$ is free for $m>2$.

so far all $\mathbb{Q}$-representations $V$ we have

$$H^k({\mathrm{SL}}_2(\mathbb{Z}),V)\simeq H^l({\mathrm{SL}}_2(\mathbb{Z}),V{)}^{{\mathrm{SL}}_2(\mathbb{Z}/m\mathbb{Z})}$$H^k(SL_2(\mathbb{Z}), V) \simeq H^l(SL_2(\mathbb{Z}), V)^{SL_2(\mathbb{Z}/m\mathbb{Z})}

due to the freeness we have also that

$$H^k({\mathrm{SL}}_2(\mathbb{Z}),V)=0$$H^k(SL_2(\mathbb{Z}), V) = 0

for $k\ge 2$

and hence

$$H^2({\mathrm{SL}}_2(\mathbb{Z}),\mathbb{Z})$$H^2(SL_2(\mathbb{Z}), \mathbb{Z})

is torsion

$$\cdots \simeq \mathrm{Hom}(H_1({\mathrm{SL}}_2(\mathbb{Z}),\mathbb{Z}),\mathbb{Q}/\mathbb{Z})\simeq \mathbb{Z}/12\mathbb{Z}.$$\cdots \simeq Hom(H_1(SL_2(\mathbb{Z}), \mathbb{Z}), \mathbb{Q}/\mathbb{Z}) \simeq \mathbb{Z}/12 \mathbb{Z} \,.

proposition

as orbifolds, we have an isomorphism

$${ℳ}_{1,1}\simeq X//(S_3\times C_2)$$\mathcal{M}_{1,1} \simeq X//(S_3 \times C_2)

where

$$X:={\mathbb{P}}^1-\{0,1,\mathrm{\infty }\}$$X := \mathbb{P}^1 - \{0,1, \infty\}

and $S_3$ acts on that by permuting $0,1,\mathrm{\infty }$. (Think of ${\mathbb{P}}^1$ as the Riemann sphere?: there is a unique holomorphic automorphism of that permuting these three points in a given fashion.) While $C_2$ acts trivially.

proof*

$$\begin{array}{ccccccc}1\to {\mathrm{SL}}_2(\mathbb{Z})[2]& \to & {\mathrm{SL}}_2(\mathbb{Z})& \to & S_3\simeq {\mathrm{SL}}_2(\mathbb{Z}/2\mathbb{Z})& \to & 1\\ & & & {}_{\varphi }\searrow & \downarrow \\ & & & & {\mathrm{PSL}}_2(\mathbb{Z})\end{array}$$\array{ 1 \to SL_2(\mathbb{Z})[2] &\to& SL_2(\mathbb{Z}) &\to& S_3 \simeq SL_2(\mathbb{Z}/2\mathbb{Z}) &\to& 1 \\ &&& {}_{\phi}\searrow& \downarrow \\ &&&& PSL_2(\mathbb{Z}) }

now ${\mathrm{PSL}}_2(\mathbb{Z})[2]$ is known to be torsion free. It acts in a standard way on the upper half plane? $𝔥$.

A little discussion shows that

$$𝔥/{\mathrm{PSL}}_2(mathbZ)[2]\simeq X$$\mathfrak{h}/PSL_2(\mathb{Z})[2] \simeq X

this implies that

$${\mathrm{PSL}}_2(\mathbb{Z})[2]\simeq F_2$$PSL_2(\mathbb{Z})[2] \simeq F_2

the free group on two generators.

Then the second but last map

$$1\to C_2\to {\mathrm{SL}}_2(\mathbb{Z})[2]\to F_2\to 1$$1 \to C_2 \to SL_2(\mathbb{Z})[2] \to F_2 \to 1

has a section, from which we get that

$${\mathrm{SL}}_2(\mathbb{Z})[2]\simeq F_2\times C_2$$SL_2(\mathbb{Z})[2] \simeq F_2 \times C_2

and so

$$X//(C_2\times S_3)\simeq (X//C_2)//S_3\simeq ((𝔥//{\mathrm{SPL}}_2(\mathbb{Z})[2])//C_2)//S_3\simeq ((𝔥//{\mathrm{PSL}}_2(\mathbb{Z}[2]))//S_3)\simeq 𝔥/{\mathrm{SL}}_2(\mathbb{Z})$$X//(C_2 \times S_3) \simeq (X//C_2) // S_3 \simeq ((\mathfrak{h}// SPL_2(\mathbb{Z})[2])//C_2)//S_3 \simeq ((\mathfrak{h}//PSL_2(\mathbb{Z}[2]))//S_3) \simeq \mathfrak{h}/ SL_2(\mathbb{Z})

which is the end of the proof.

corollary The Euler characteristic of the moduli stack of elliptic curves is

$$\chi ({ℳ}_{1,1})=\frac{-1}{12}.$$\chi(\mathcal{M}_{1,1}) = \frac{-1}{12} \,.

now consider the line bundle

$$\begin{array}{c}ℂ\times 𝔥\\ \downarrow \\ 𝔥//{\mathrm{SL}}_2(\mathbb{Z})\end{array}$$\array{ \mathbb{C} \times \mathfrak{h} \\ \downarrow \\ \mathfrak{h}//SL_2(\mathbb{Z}) }

with action on the total space for $k\in \mathbb{Z}$

$$\left(\begin{array}{cc}a& b\\ c& d\end{array}\right):(z,\tau )\mapsto (c\tau +d{)}^{k}z,\frac{a\tau +b}{c\tau +d}$$\left( \array{ a & b \\ c & d } \right) : (z, \tau) \mapsto (c \tau + d)^k z, \frac{a \tau + b}{c \tau + d}

call this line bundle on the moduli stack ${ℒ}_k\to {ℳ}_{1,1}$. We will see that all line bundles are isomorphic to one of these.

remark

$$f:𝔥\to 𝒞$$f : \mathfrak{h} \to \mathcal{C}

is a section of ${ℒ}_k$ iff

$$f\left(\frac{a\tau +b}{c\tau +d}\right)=(c\tau +d{)}^{k}f(\tau )$$f\left( \frac{a \tau + b}{c \tau + d} \right) = (c \tau + d)^k f(\tau)

hence precisely if it defines a modular function of weight $k$! This gives a geometric interpretation of modular functions.

$$\begin{array}{c}ℂ\times 𝔥\\ \downarrow \\ 𝔥\end{array}$$\array{ \mathbb{C} \times \mathfrak{h} \\ \downarrow \\ \mathfrak{h} }

and define an action of $G:={\mathrm{SL}}_2(\mathbb{Z})letimes{\mathbb{Z}}^2$

where ${\mathbb{Z}}^2$ acts on ${\mathrm{SL}}_2(\mathbb{Z})$ by

$$\left(\begin{array}{cc}a& b\\ c& d\end{array}\right):(m,n)\mapsto am+bm,cm+dn$$\left( \array{ a & b \\ c & d } \right) : (m , n) \mapsto a m + b m, c m + d n

and on $ℂ\times 𝔥$ by

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

the resulting bundle

$$\begin{array}{c}ℂ\times 𝔥//G\\ \downarrow \\ {ℳ}_{1,1}\end{array}$$\array{ \mathbb{C} \times \mathfrak{h}//G \\ \downarrow \\ \mathcal{M}_{1,1} }

we call

$$ℰ\to {ℳ}_{1,1}$$\mathcal{E} \to \mathcal{M}_{1,1}

theorem for any complex manifold $T$ there is a bijection between families of elliptic curves over $T$ and orbifold maps $T\to {ℳ}_{1,1}$ classify them.

Suppose we have an “isotrivial family” (meaning all fibers are isomorphic elliptic curves, i.e. a fiber bundle of elliptic curves)

$$\begin{array}{c}\\ \downarrow \\ T& \stackrel{\varphi }{\to }& {ℳ}_{1,1}\end{array}$$\array{ \\ \downarrow \\ T &\stackrel{\phi}{\to}& \mathcal{M}_{1,1} }

recall that the group that defines $T$ as an orbifold is the first homotopy group ${\pi }_1(T)$.

The only condition that we get from the definition of orbifold maps is that

$$\varphi :{\pi }_1(T)\to {\mathrm{SL}}_2(\mathbb{Z})$$\phi : \pi_1(T) \to SL_2(\mathbb{Z})

factors through the stabilizer group $\simeq \mathrm{Aut}(E_p)$ of our base point $p\in {ℳ}_{1,1}$

compactified moduli stack

one can see that over compact $T$ with ${ℳ}_{1,1}$ we cannot have nontrivial famlies without singular fibers.

To get around that we want a compactification ${\overline{ℳ}}_{1,1}$ of the moduli stack.

also fur purposes of intersection theory, we need to further compactify.

recall the description of ${ℳ}_{1,1}$ as a weak quotient of ${\mathbb{P}}^1$. Then consider:

definition

Let

$${\overline{ℳ}}_{1,1}:={\mathbb{P}}^1//(C_2\times S_3)$$\bar \mathcal{M}_{1,1} := \mathbb{P}^1//(C_2 \times S_3)

otice that this is now an orbifold which is no longer basic by the above definition. In fact, we can cover it by charts of basic orbifolds as follows: consider

$$\begin{array}{ccc}& & 𝔥//(C_2\times \mathbb{Z})\\ & \swarrow & & \searrow \\ 𝔥//{\mathrm{SL}}_2(\mathbb{Z})& & & & 𝔻//C_2\end{array}$$\array{ && \mathfrak{h}//(C_2 \times \mathbb{Z}) \\ & \swarrow && \searrow \\ \mathfrak{h}//SL_2(\mathbb{Z}) &&&& \mathbb{D}//C_2 }

with the arrows being maps of orbifolds whose precise details I haven’t typed.

then let ${𝔻}^{*}$ be the punctured disk and realize the diagram

$$\begin{array}{ccc}& & {𝔻}^{*}//C_2\\ & \swarrow & & \searrow \\ {ℳ}_{1,1}& & & & 𝔻//C_2\end{array}$$\array{ && \mathbb{D}^*//C_2 \\ & \swarrow && \searrow \\ \mathcal{M}_{1,1} &&&& \mathbb{D}//C_2 }

where the right morphism is just the inclusion

now we build a chart of ${\overline{ℳ}}_{1,1}$ consisting of the two patches ${ℳ}_{1,1}$ and $𝔻/C_2/$

from this we get the alternative

definition

$${\overline{ℳ}}_{1,1}:={ℳ}_{1,1}\coprod _{{𝔻}^{*}//C_2}𝔻//C_2$$\bar \mathcal{M}_{1,1} := \mathcal{M}_{1,1} \coprod_{\mathbb{D}^*//C_2} \mathbb{D}//C_2

the colimit on the right manifestly glues in the “point at infinity” that is not hit by the map ${𝔻}^{*}//C_2\to {ℳ}_{1,1}$.

Gromov-Witten invariants

definition A stable curve (over $𝒞$) of genus $g$ with $n$ marked points is a proper, connected curve with $n$ smooth marked points such that all singularities are nodes and such that the the automorphism group (of autos respecting the smooth marked points) is finite,

$$\mid \mathrm{Aut}(C)\mid <+\mathrm{\infty }$$|Aut(C)| \lt + \infty

and such that the arithmetic genus? is $g$.

Now $bat{ℳ}_{g,n}$ is the fine moduli space for smooth curves of genus $g$.

There is a line bundle

$${𝒯}_i^{*}\to {\overline{ℳ}}_{g,n}$$\mathcal{T}_i^* \to \bar \mathcal{M}_{g,n}

built fiberwise from the cotangent spaces of the elliptic curves.

one of them is obtained from one of the $n$ sections $s_i$ of the universal family $ℱ\to {\overline{ℳ}}_{g,n}$. The fiber over a point is the cotangent space of the elliptic curve over that point at this section.

Write for the first Chern class

$$c_1({𝒯}_i^{*})={\Psi }_i$$c_1(\mathcal{T}_i^*) = \Psi_i

$$k_1,\cdots ,k_n\in {\mathbb{Z}}_{\ge 0}$$k_1, \cdots, k_n \in \mathbb{Z}_{\geq 0}

such that

$$\sum _{i=1}^n{k}_i=3g-3+n$$\sum_{i = 1}^n k_i = 3 g - 3 + n

then we get numbers called the Gromov-Witten invariants (“of the point”)

$$⟨{\tau }_{k_1},\cdots ,{\tau }_{k_n}{⟩}_g:={\int }_{{\overline{ℳ}}_{1,1}}\prod _{i=1}^n{\Psi }_i^{k_i}$$\langle \tau_{k_1}, \cdots, \tau_{k_n} \rangle_g := \int_{\bar \mathcal{M}_{1,1}} \prod_{i = 1}^n \Psi_i^{k_i}

example: $⟨{\tau }_1{⟩}_1$

Let $x,y$ by affine coordinates on ${\mathbb{P}}^2$

Let $f(x,y)$ and $g(x,y)$ be two generic cubics, in particular there are nine joint zeros

$$\mid ⟨(x,y)\mid f(x,y)=g(x,y)=0⟩\mid =0$$|\langle (x,y)| f(x,y) = g(x,y) = 0\rangle| = 0

called $p_1,\cdots ,p_9$.

define then

$$F:=\{(x,y,t)\in {\mathbb{P}}^2\times {\mathbb{P}}^1:f(x,y)-tg(x,y)=0\}$$F := \left\lbrace (x,y,t) \in \mathbb{P}^2 \times \mathbb{P}^1 : f(x,y) - t g(x,y) = 0 \right\rbrace

and consider

$$\begin{array}{c}F\\ {\downarrow }^{{\mathrm{pr}}_2}\\ {\mathbb{P}}^1& \stackrel{\varphi }{\to }& {\overline{ℳ}}_{1,1}\\ & \searrow & {\downarrow }^q\\ & & {\overline{M}}_{1,1}\end{array}$$\array{ F \\ \downarrow^{pr_2} \\ \mathbb{P}^1 &\stackrel{\phi}{\to}& \bar \mathcal{M}_{1,1} \\ &\searrow & \downarrow^{q} \\ && \bar M_{1,1} }

That map $q$ has degree $\frac{1}{2}$ (!) since ${\mathbb{P}}^1\to {\overline{ℳ}}_{1,1}$ has degree 12

we also find that the diaginal map ${\mathbb{P}}^1\to {\overline{M}}_{1,1}$ has degree 12. It follows that $\varphi$ has degree 24:

$$\mathrm{deg}(\varphi )=24.$$deg(\phi) = 24 \,.

Now let ${𝒯}_i^{*}\to {\overline{ℳ}}_{g,n}$ be one of these line bundles. Consider the pullback ${\varphi }^{*}({𝒯}_1)$

then by some argument not reproduced here we find

$${\int }_{{\mathbb{P}}^1}{c}_1({\varphi }^{*}({𝒯}_1{)}^{*}).$$\int_{\mathbb{P}^1} c_1(\phi^*(\mathcal{T}_1)^*) \,.

Then since the order of $\varphi$ is 24 we find that the first Gromov-Witten invariant is

$$⟨{\tau }_1{⟩}_1=\frac{1}{24}.$$\langle \tau_1 \rangle_1 = \frac{1}{24} \,.

extending structures to the compactified moduli space

recall that the moduli stack of elliptic curve is, as a global orbifold

$${ℳ}_{1,1}:=𝔥//{\mathrm{SL}}_2(\mathbb{Z})$$\mathcal{M}_{1,1} := \mathfrak{h}//SL_2(\mathbb{Z})

and also

$$\cdots \simeq ({\mathbb{P}}^1-\{0,1,\mathrm{\infty }\})//(C_2\times S_3)$$\cdots \simeq (\mathbb{P}^1 - \{0,1,\infty\})//(C_2 \times S_3)

and there is a line bundle on this given by

$${ℒ}_k:=(ℂ\times 𝔥)//{\mathrm{SL}}_2(\mathbb{Z})$$\mathcal{L}_k := (\mathbb{C} \times \mathfrak{h})//SL_2(\mathbb{Z})

where the action is given by

$$\left(\begin{array}{cc}a& b\\ c& d\end{array}\right):(z,\tau )\mapsto (c\tau +d{)}^{k}z,\frac{a\tau +b}{c\tau +d}$$\left( \array{ a& b \\ c & d } \right) : (z, \tau) \mapsto (c\tau + d)^{k} z, \frac{a \tau + b}{c \tau + d}

since ${\mathrm{SL}}_2(\mathbb{Z}{)}^{\mathrm{ab}}=\mathbb{Z}/12\mathbb{Z}$ one finds the Picard group

$$\mathrm{Pic}{ℳ}_{1,1}\simeq \mathbb{Z}/12\mathbb{Z}$$Pic \mathcal{M}_{1,1} \simeq \mathbb{Z}/12\mathbb{Z}

meromorphic (holomorphic) sections $f$ of ${ℒ}_k$ are modular functions of weight $k$, i.e. $f:𝔥\to ℂ$ such that

$$\forall \gamma =(\frac{ab}{cd})\in {\mathrm{SL}}_2(\mathbb{Z}):f(\gamma \tau )=(c\tau +d{)}^{k}f(\tau )$$\forall \gamma = (\frac{a b}{c d}) \in SL_2(\mathbb{Z}) : f(\gamma \tau) = (c \tau + d)^k f(\tau)

the universal elliptic curve over ${ℳ}_{1,1}$ is

$$ℰ:=(ℂ\times 𝔥)//({\mathrm{SL}}_2(\mathbb{Z})⋉{\mathbb{Z}}^2)$$\mathcal{E} := (\mathbb{C} \times \mathfrak{h})//(SL_2(\mathbb{Z}) \ltimes \mathbb{Z}^2)

Then we ended last time with describing the compactified moduli space

$${\overline{ℳ}}_{1,1}:={\mathbb{P}}^1//(C_2\times S_3)$$\bar \mathcal{M}_{1,1} := \mathbb{P}^1//(C_2 \times S_3)

extending the line bundles

proposition ${𝕃}_k$ has a universal extension ${\overline{ℒ}}_k$ to ${\overline{ℳ}}_{1,1}$

proof

take

$$(ℂ\times 𝔻//C_2)\to (𝔻//C_2)$$(\mathbb{C} \times \mathbb{D}//C_2) \to (\mathbb{D}//C_2)

where $C_2$ acts by

$$\pm 1(z,\tau )\mapsto ({\pm }^{k}z,\tau )$$\pm 1 \;\; (z,\tau) \mapsto (\pm^k z , \tau)

note that since $\left(\begin{array}{cc}1& n\\ 01\end{array}\right)\in S_2(\mathbb{Z})$ for all $n\in \mathbb{Z}$ ay modular function $f:𝔥\to ℂ$

$$f(\tau )=\sum {-\mathrm{\infty }}^{\mathrm{\infty }}{a}_n{q}^n$$f(\tau) = \sum{-\infty}^\infty a_n q^ n

where $q:=e^{2\pi i\tau }$

is called a holomorphic modular form of weight $k$ if $f:𝔥\to ℂ$ is holomorphic and $a_n=0$ for all $n<0$

remark modular forms of weight $k$ are in bijection with sections of the line bundle ${\overline{ℒ}}_k$.

example for any lattice $\Lambda$ in $ℂ$ and for any $k>2$ we have

$$S_k(\Lambda ):={Sum}_{0\ne \lambda \in \Lambda }\frac{1}{{\lambda }^k}$$S_k(\Lambda) := \Sum_{0 \neq \lambda \in \Lambda} \frac{1}{\lambda^k}

obviously for all $u\in {ℂ}^{*}$ $S_k(u\Lambda )=u^{-k}{S}_k(\Lambda )$

and for all $\tau \in 𝔥$ with $C_k(\tau ):=S_k({\Lambda }_{\tau })$ it follows that $G_k:𝔥\to ℂ$ is holomorphic

since ${\Lambda }_{\gamma \tau }=(c\tau +d{)}^{-1}{\Lambda }_{\tau }$ it follows that $G_k$ is a modular function of weight $k$

fact $G_{2k}=2\zeta (2k)\text{Unknown character}\text{Unknown character}2\frac{(2\pi i{)}^{2k}}{(2k-1)!}{\sum }_{n=1}^{\mathrm{\infty }}{b}_{2k-1}(n)q^n$

with $b_k(n):={\sum }_{d\mid n}{d}^k$ and where $\zeta$ is the zeta-function?

it follows that

$G_{2k}$ is a modular form of weight $2k$ (which is not a cusp form).

an important cusp form is

setting $g_2=60G_4$ and $g_3=140G_6$

the modular form

$$\Delta :=g_2(\tau {)}^3-27g_3(\tau {)}^2$$\Delta := g_2(\tau)^3 - 27 g_3(\tau)^2

is a cusp form of weight 12. $\Delta$ does not have any 0 in $𝔥$ and it has a simple zero at $q=0$.

we have an isomorphism

$${\overline{ℒ}}_{12}\simeq {𝒪}_{{\overline{ℳ}}_{1,1}(\mathrm{\infty })}$$\bar \mathcal{L}_{12} \simeq \mathcal{O}_{\bar \mathcal{M}_{1,1}(\infty)}

where on the right is the sheaf with at most a pole at $\mathrm{\infty }$. This isomorphism going from right to left is induced by multiplication with $\Delta$.

we have an exact sequence

$$0\to \mathbb{Z}\to \mathrm{Pic}({\overline{ℳ}}_{1,1})\to \mathbb{Z}/12\mathbb{Z}\to 0$$0 \to \mathbb{Z} \to Pic(\bar \mathcal{M}_{1,1}) \to \mathbb{Z}/12\mathbb{Z} \to 0

where the first nontrivial map sends 1 to ${\overline{ℒ}}_{12}$ and the second one ${\overline{ℒ}}_1$ to the generator.

set for all $k$

$$M_k:=\{\mathrm{modular}\mathrm{forms}\mathrm{of}\mathrm{weight}k\}$$M_k := \{modular forms of weight k\}

$$M_k^{\circ }:=\{\mathrm{cusp}\mathrm{forms}\mathrm{of}\mathrm{weight}k\}$$M_k^\circ := \{cusp forms of weight k\}

proposition $M_{•}$ is an even graded algebra freely generated by $G_4$ and $G_6$ and the ideal $M_{•}^{\circ }$ is generated by $\Delta$.

the dimensions are

dim M_{2k} = \left\{ floor k/6 & for k = 1 mod 6 \\ 1+ floor k/6 & otherwise \right. dim M_{2k} = \left\{ floor k/6 & for k = 1 mod 6 \\ 1+ floor k/6 & otherwise \right.

???????

extending the universal family of elliptic curves

Recall the three definition of elliptic curves from above.

Now a fourth definition:

definition an elliptic curve is a smooth curve of degree 3 in $ℂ{\mathbb{P}}^2$ together with a point in it.

• that this equation implies the first one above follows from the genus formula, which says that a degree $n$ curve as in the definition has genus $g=\frac{(n-1)(n-2)}{2}$

• that the first def implies this one