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A Survey of Elliptic Cohomology - elliptic curves

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This is a sub-entry of

and

see there for background and context.

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

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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


Contents

elliptic curves

Definition An elliptic curve over \mathbb{C} is equivalently

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 \mathbb{C}.

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

A framed lattice in \mathbb{C} is a lattice Λ\Lambda together with an ordered basis (λ 1,λ 2)(\lambda_1, \lambda_2) of Λ\Lambda such that Im(λ 2/λ 1)>0Im(\lambda_2/\lambda_1) \gt 0.

(So turning λ 1\lambda_1 to λ 2\lambda_2 in the plane means going counterclockwise).

moduli spaces of elliptic curves

this implies that

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

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

and we have

{ellipticcurvesover} 𝔥/SL 2() \{elliptic curves over \mathbb{C}\}_\sim \simeq \mathfrak{h}/{SL_2(\mathbb{Z})}

where SL 2()={(a b c d)adcd=1}SL_2(\mathbb{Z}) = \left\{ \left(\array{a & b \\ c & d }\right)| a d - c d = 1\right\} acts by

τaτ+bcτ+d \tau \mapsto \frac{a \tau + b}{c \tau + d}

Claim the quotient 𝔥/ SL 2()\mathfrak{h}/_{SL_2(\mathbb{Z})} is biholomorphic? to the disk and has a unique structure of a Riemann surface which makes the quotient map 𝔥𝔥/SL 2()\mathfrak{h} \to \mathfrak{h}/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/SL 2()M_{1,1} := \mathfrak{h}/SL_2(\mathbb{Z})

definition a homolorphic family of elliptic curves over a complex manifold TT is

  • a holomorphic map π:XT\pi : X \to T

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

For every family

X π T \array{ X \\ \downarrow^\pi \\ T }

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

Xϕ *F F T M 1,1 \array{ X \simeq \phi^* F &\to& F \\ \downarrow & & \downarrow \\ T &\to& M_{1,1} }

where

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

such that

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

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

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

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

For instance

{(x,y,z) 2×X:y 2=x(x1)(xλ)}X:= 1{0,1,} \{ (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

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

given by

(m,n):(z,τ)(z+mτ+n,τ) (m,n) : (z,\tau) \mapsto (z + m \tau + n, \tau)

Then consider the family

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

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

and E τ=/Λ τE_\tau = \mathbb{C}/{\Lambda_\tau} with

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

is a family of framed elliptic curves.

fact the space 𝔥\mathfrak{h} with the family E𝔥E \to \mathfrak{h} is a fine moduli space for framed elliptic curves.

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

with pullback of the universal family

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

claim for every point tTt \in T there is an open neighbourhood t 0UTt_0 \in U \hookrightarrow T such that one can choose 1-forms ω t\omega_t on X τX_\tau which vary holomorphically with respect to tt.

Notice that locally every family of elliptic curves is framed (since we can locally extend a choice of basis for H 1H_1). So

𝔥 SL 2() M 1,1 Id M 1,1 \array{ && \mathfrak{h} \\ && \downarrow^{SL_2(\mathbb{Z})} \\ M_{1,1} &\stackrel{Id}{\to}& M_{1,1} }

at ii and ρ=e 2πi/6\rho = e^{2\pi i/6} , C={±I}C = \{\pm I\}

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

orbifolds

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

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

A morphism from (X,γ,ρ)(X,\gamma, \rho) to (X,Γ,ρ)(X', \Gamma', \rho') is a pair

(f,ϕ) (f,\phi)

where

  • f:XXf : X \to X' is a continuous map

  • ϕ:ΓΓ\phi : \Gamma \to \Gamma' is a group homomorphism

such that for all γΓ\gamma \in \Gamma

X f X γ ϕ(γ) X f X \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 XX with universal cover X˜\tilde X we have

XX˜//π 1(X) X \mapsto \tilde X //\pi_1(X)

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

remark let XX be a nice topological space. Let G=π 1(X)G = \pi_1(X) be its first homotopy group and let a discrete group Γ\Gamma action on XX.

then define

Γ˜:={(γ,g)Γ×Aut(X˜)X˜ g X˜ p p X γ X} \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

1GΓ˜Γ1 1 \to G \to \tilde \Gamma \to \Gamma \to 1

where GΓ˜G \to \tilde \Gamma is given by (g(1,g))(g \mapsto (1,g)) and Γ˜Γ\tilde \Gamma \to \Gamma by (γ,g)γ(\gamma,g) \mapsto \gamma.

definition

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

Then a homotopy from (f,ϕ)(f,\phi) to (f,ϕ):(X,Γ,ρ)(X,Γ,ρ)(f',\phi') : (X,\Gamma, \rho) \to (X',\Gamma', \rho')

is a map

(F,Ψ):I×(X,Γ,ρ)(X,Γ,ρ) (F,\Psi) : I \times (X, \Gamma, \rho) \to (X', \Gamma', \rho')

such that

  • Ψ=ϕ=ϕ\Psi = \phi = \phi'

  • f()=F(0,)f(-) = F(0,-), f()=F(1,)f'(-) = F(1,-)

now write

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

(the circle regarded as a global orbifold)

definition

The first homotopy group for our definition of orbifold is:

π 1(X//γ){homotopyclassesofmapsS 1X//Γ} \pi_1(X//\gamma) \simeq \{ homotopy classes of maps S^1 \to X//\Gamma \}

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

(recall again the simply-connectness assumoption!!)

definition* A morphism

(f,ϕ):(X,Γ)(X,Γ) (f,\phi) : (X,\Gamma) \to (X',\Gamma')

is a weak homotopy equivalence if ϕ\phi is an ismorphism and H (f):H (X)X (X)H_\bullet(f) : H_\bullet(X) \to X_\bullet(X').

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

(EΓ×X,Γ)(f,ϕ)(X,Γ) (E \Gamma \times X, \Gamma) \stackrel{(f,\phi)}{\to} (X,\Gamma)

with ϕ=Id Γ\phi = Id_\Gamma and ff the projection is a weak homotopy equivalence.

definition a local system VV on (X,Γ)(X,\Gamma) with fiber VV a group homomorphism ΓAut(V)\Gamma \to Aut(V) with

definition

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

  • π n(X//Γ)\pi_n(X//\Gamma) := \pi_n(X)for for n \geq 2$

  • H (X//Γ,V):=H (EΓ× GamaX,V)H_\bullet(X//\Gamma, V) := H_\bullet(E \Gamma \times_\Gama X, V)

  • H (X//Γ,V):=H (EΓ× GamaX,V)H^\bullet(X//\Gamma, V) := H^\bullet(E \Gamma \times_\Gama X, V)

example *//Γ{*}//\Gamma has a weak homotopy equivalence to the classifying space Γ\mathcal{B}\Gamma

it follows that for local system VV we have

H (*//Γ,V)=H gp (Γ,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 XX with suitable extension of the action of Γ\Gamma to them

A vector bundle on an orbifold VX//ΓV \to X//\Gamma is a vector bundle VXV \to X with isomorphism action by Γ\Gamma specified, covering that on XX.

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

definition say that Γ\Gamma acts virtually freely if \exists a finite index subgroup Γ\Gamma' of Γ\Gamma which acts freely on XX.

note SL 2()SL_2(\mathbb{Z}) acts virtually freely on 𝔥\mathfrak{h}

SL 2()[m]SL 2()SL 2(/m) SL_2(\mathbb{Z})[m] \to SL_2(\mathbb{Z}) \to SL_2(\mathbb{Z}/ m \mathbb{Z})

Let Γ<Γ\Gamma' \lt \Gamma be a finite index subgroup which acts freely on XX.

set

X:=X//Γ; X' := X// \Gamma';

the map

X/ΓX//Γ X/\Gamma' \to X//\Gamma

must be viewed as an unramified covering of degree [Γ:Γ][\Gamma:\Gamma'].

supposedly important statement

definition

the Euler characteristic of a global orbifold is

χ(X//Γ):=1[Γ:Γ]χ(X/Γ) \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

1,1:=𝔥//SL 2() \mathcal{M}_{1,1} := \mathfrak{h}//SL_2(\mathbb{Z})

proposition

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

and similarly for integral cohomology

χ( 1,1)=112 \chi(\mathcal{M}_{1,1}) = -\frac{1}{12}
Pic( 1,1)/12 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// 2\mathfrak{h}//\mathbb{Z}_2 are the same as that of *// 2* // \mathbb{Z}_2 and similarly for the (group)cohomology

H ( 1,1,)=H(SL 2(),) H^\bullet(\mathcal{M}_{1,1}, \mathbb{Z}) = H\bullet(SL_2(\mathbb{Z}), \mathbb{Z})

and similalry for homology.

In particular

H 1( 1,1,)SL 2() ab/12 H_1(\mathcal{M}_{1,1}, \mathbb{Z}) \simeq SL_2(\mathbb{Z})^{ab} \simeq \mathbb{Z}/12\mathbb{Z}

for all mm \in \mathbb{N} then

1SL 2()[m]SL 2()SL 2(/m)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( 1,1,)=0 H^1(\mathcal{M}_{1,1}, \mathbb{Z}) = 0

fact the group SL 2()[m]SL_2(\mathbb{Z})[m] is free for m>2m \gt 2.

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

H k(SL 2(),V)H l(SL 2(),V) SL 2(/m) 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(SL 2(),V)=0 H^k(SL_2(\mathbb{Z}), V) = 0

for k2k \geq 2

and hence

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

is torsion

Hom(H 1(SL 2(),),/)/12. \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

1,1X//(S 3×C 2) \mathcal{M}_{1,1} \simeq X//(S_3 \times C_2)

where

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

and S 3S_3 acts on that by permuting 0,1,0,1, \infty. (Think of 1\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 2C_2 acts trivially.

proof*

1SL 2()[2] SL 2() S 3SL 2(/2) 1 ϕ PSL 2() \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 PSL 2()[2]PSL_2(\mathbb{Z})[2] is known to be torsion free. It acts in a standard way on the upper half plane 𝔥\mathfrak{h}.

A little discussion shows that

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

this implies that

PSL 2()[2]F 2 PSL_2(\mathbb{Z})[2] \simeq F_2

the free group on two generators.

Then the second but last map

1C 2SL 2()[2]F 21 1 \to C_2 \to SL_2(\mathbb{Z})[2] \to F_2 \to 1

has a section, from which we get that

SL 2()[2]F 2×C 2 SL_2(\mathbb{Z})[2] \simeq F_2 \times C_2

and so

X//(C 2×S 3)(X//C 2)//S 3((𝔥//SPL 2()[2])//C 2)//S 3((𝔥//PSL 2([2]))//S 3)𝔥/SL 2() 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

χ( 1,1)=112. \chi(\mathcal{M}_{1,1}) = \frac{-1}{12} \,.

now consider the line bundle

×𝔥 𝔥//SL 2() \array{ \mathbb{C} \times \mathfrak{h} \\ \downarrow \\ \mathfrak{h}//SL_2(\mathbb{Z}) }

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

(a b c d):(z,τ)(cτ+d) kz,aτ+bcτ+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 k 1,1\mathcal{L}_k \to \mathcal{M}_{1,1}. We will see that all line bundles are isomorphic to one of these.

remark

f:𝔥𝒞 f : \mathfrak{h} \to \mathcal{C}

is a section of k\mathcal{L}_k iff

f(aτ+bcτ+d)=(cτ+d) kf(τ) 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 kk! This gives a geometric interpretation of modular functions.

×𝔥 𝔥 \array{ \mathbb{C} \times \mathfrak{h} \\ \downarrow \\ \mathfrak{h} }

and define an action of G:=SL 2()letimes 2G := SL_2(\mathbb{Z}) \letimes \mathbb{Z}^2

where 2\mathbb{Z}^2 acts on SL 2()SL_2(\mathbb{Z}) by

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

and on ×𝔥\mathbb{C} \times \mathfrak{h} by

(m,n):(z,τ)(z+mτ+n,τ) (m, n) : (z,\tau) \mapsto (z + m \tau + n, \tau)

the resulting bundle

×𝔥//G 1,1 \array{ \mathbb{C} \times \mathfrak{h}//G \\ \downarrow \\ \mathcal{M}_{1,1} }

we call

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

theorem for any complex manifold TT there is a bijection between families of elliptic curves over TT and orbifold maps T 1,1T \to \mathcal{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)

T ϕ 1,1 \array{ \\ \downarrow \\ T &\stackrel{\phi}{\to}& \mathcal{M}_{1,1} }

recall that the group that defines TT as an orbifold is the first homotopy group π 1(T)\pi_1(T).

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

ϕ:π 1(T)SL 2() \phi : \pi_1(T) \to SL_2(\mathbb{Z})

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

compactified moduli stack

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

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

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

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

definition

Let

¯ 1,1:= 1//(C 2×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

𝔥//(C 2×) 𝔥//SL 2() 𝔻//C 2 \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 *\mathbb{D}^* be the punctured disk and realize the diagram

𝔻 *//C 2 1,1 𝔻//C 2 \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 ¯ 1,1\bar \mathcal{M}_{1,1} consisting of the two patches 1,1\mathcal{M}_{1,1} and 𝔻/C 2/\mathbb{D}/C_2/

from this we get the alternative

definition

¯ 1,1:= 1,1 𝔻 *//C 2𝔻//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 1,1\mathbb{D}^*//C_2 \to \mathcal{M}_{1,1}.

Gromov-Witten invariants

definition A stable curve (over 𝒞\mathcal{C}) of genus gg with nn marked points is a proper, connected curve with nn 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,

Aut(C)<+ |Aut(C)| \lt + \infty

and such that the arithmetic genus is gg.

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

There is a line bundle

𝒯 i *¯ 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 nn sections s is_i of the universal family ¯ g,n\mathcal{F} \to \bar \mathcal{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 *)=Ψ i c_1(\mathcal{T}_i^*) = \Psi_i
k 1,,k n 0 k_1, \cdots, k_n \in \mathbb{Z}_{\geq 0}

such that

i=1 nk i=3g3+n \sum_{i = 1}^n k_i = 3 g - 3 + n

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

τ k 1,,τ k n g:= ¯ 1,1 i=1 nΨ 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: τ 1 1\langle \tau_1\rangle_1

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

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

(x,y)f(x,y)=g(x,y)=0=0 |\langle (x,y)| f(x,y) = g(x,y) = 0\rangle| = 0

called p 1,,p 9p_1, \cdots, p_9.

define then

F:={(x,y,t) 2× 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

F pr 2 1 ϕ ¯ 1,1 q M¯ 1,1 \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 qq has degree 12\frac{1}{2} (!) since 1¯ 1,1\mathbb{P}^1 \to \bar \mathcal{M}_{1,1} has degree 12

we also find that the diaginal map 1M¯ 1,1\mathbb{P}^1 \to \bar M_{1,1} has degree 12. It follows that ϕ\phi has degree 24:

deg(ϕ)=24. deg(\phi) = 24 \,.

Now let 𝒯 i *¯ g,n\mathcal{T}_i^* \to \bar \mathcal{M}_{g,n} be one of these line bundles. Consider the pullback ϕ *(𝒯 1)\phi^*(\mathcal{T}_1)

then by some argument not reproduced here we find

1c 1(ϕ *(𝒯 1) *). \int_{\mathbb{P}^1} c_1(\phi^*(\mathcal{T}_1)^*) \,.

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

τ 1 1=124. \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

1,1:=𝔥//SL 2() \mathcal{M}_{1,1} := \mathfrak{h}//SL_2(\mathbb{Z})

and also

( 1{0,1,})//(C 2×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

k:=(×𝔥)//SL 2() \mathcal{L}_k := (\mathbb{C} \times \mathfrak{h})//SL_2(\mathbb{Z})

where the action is given by

(a b c d):(z,τ)(cτ+d) kz,aτ+bcτ+d \left( \array{ a& b \\ c & d } \right) : (z, \tau) \mapsto (c\tau + d)^{k} z, \frac{a \tau + b}{c \tau + d}

since SL 2() ab=/12SL_2(\mathbb{Z})^{ab} = \mathbb{Z}/12\mathbb{Z} one finds the Picard group

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

meromorphic (holomorphic) sections ff of k\mathcal{L}_k are modular functions of weight kk, i.e. f:𝔥f : \mathfrak{h} \to \mathbb{C} such that

γ=(abcd)SL 2():f(γτ)=(cτ+d) kf(τ) \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 1,1\mathcal{M}_{1,1} is

:=(×𝔥)//(SL 2() 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

¯ 1,1:= 1//(C 2×S 3) \bar \mathcal{M}_{1,1} := \mathbb{P}^1//(C_2 \times S_3)

extending the line bundles

proposition 𝕃 k\mathbb{L}_k has a universal extension ¯ k\bar \mathcal{L}_k to ¯ 1,1\bar \mathcal{M}_{1,1}

proof

take

(×𝔻//C 2)(𝔻//C 2) (\mathbb{C} \times \mathbb{D}//C_2) \to (\mathbb{D}//C_2)

where C 2C_2 acts by

±1(z,τ)(± kz,τ) \pm 1 \;\; (z,\tau) \mapsto (\pm^k z , \tau)

note that since (1 n 01)S 2()\left(\array{1 & n \\ 0 1}\right) \in S_2(\mathbb{Z}) for all nn \in \mathbb{Z} ay modular function f:𝔥f : \mathfrak{h} \to \mathbb{C}

f(τ)= a nq n f(\tau) = \sum{-\infty}^\infty a_n q^ n

where q:=e 2πiτq := e^{2 \pi i \tau}

is called a holomorphic modular form of weight kk if f:𝔥f : \mathfrak{h} \to \mathbb{C} is holomorphic and a n=0a_n = 0 for all n<0n \lt 0

remark modular forms of weight kk are in bijection with sections of the line bundle ¯ k\bar \mathcal{L}_k.

example for any lattice Λ\Lambda in \mathbb{C} and for any k>2k \gt 2 we have

S k(Λ):=Sum 0λΛ1λ k S_k(\Lambda) := \Sum_{0 \neq \lambda \in \Lambda} \frac{1}{\lambda^k}

obviously for all u *u \in \mathbb{C}^* S k(uΛ)=u kS k(Λ)S_k(u \Lambda) = u^{-k} S_k(\Lambda)

and for all τ𝔥\tau \in \mathfrak{h} with C k(τ):=S k(Λ τ)C_k(\tau) := S_k(\Lambda_\tau) it follows that G k:𝔥G_k : \mathfrak{h} \to \mathbb{C} is holomorphic

since Λ γτ=(cτ+d) 1Λ τ\Lambda_{\gamma \tau} = (c \tau + d)^{-1} \Lambda_\tau it follows that G kG_k is a modular function of weight kk

fact G 2k=2ζ(2k)Unknown characterUnknown character2(2πi) 2k(2k1)! n=1 b 2k1(n)q nG_{2k} = 2 \zeta(2 k) ü 2 \frac{(2 \pi i)^{2k}}{(2k-1)!} \sum_{n=1}^\infty b_{2k-1}(n)q^n

with b k(n):= dnd kb_k(n) := \sum_{d|n} d^k and where ζ\zeta is the zeta-function?

it follows that

G 2kG_{2k} is a modular form of weight 2k2k (which is not a cusp form).

an important cusp form is

setting g 2=60G 4g_2 = 60 G_4 and g 3=140G 6g_3 = 140 G_6

the modular form

Δ:=g 2(τ) 327g 3(τ) 2 \Delta := g_2(\tau)^3 - 27 g_3(\tau)^2

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

we have an isomorphism

¯ 12𝒪 ¯ 1,1() \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 \infty. This isomorphism going from right to left is induced by multiplication with Δ\Delta.

we have an exact sequence

0Pic(¯ 1,1)/120 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 ¯ 12\bar \mathcal{L}_{12} and the second one ¯ 1\bar \mathcal{L}_1 to the generator.

set for all kk

M k:={modularformsofweightk} M_k := \{modular forms of weight k\}
M k :={cuspformsofweightk} M_k^\circ := \{cusp forms of weight k\}

proposition M M_\bullet is an even graded algebra freely generated by G 4G_4 and G 6G_6 and the ideal M M_\bullet^\circ is generated by Δ\Delta.

the dimensions are

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 2 dim M_{2k} = \left\{ floor k/6 &\mathbb{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 nn curve as in the definition has genus g=(n1)(n2)2g = \frac{(n-1)(n-2)}{2}

  • that the first def implies this one

Revised on November 6, 2013 00:47:48 by Urs Schreiber (77.251.114.72)