Moduli stack of elliptic curves explained

In mathematics, the moduli stack of elliptic curves, denoted as

l{M}_1,1

l{M}_ell

Spec(Z)

classifying elliptic curves. Note that it is a special case of the moduli stack of algebraic curves

l{M}_g,n

. In particular its points with values in some field correspond to elliptic curves over the field, and more generally morphisms from a scheme

to it correspond to elliptic curves over

. The construction of this space spans over a century because of the various generalizations of elliptic curves as the field has developed. All of these generalizations are contained in

l{M}_1,1

Properties

Smooth Deligne-Mumford stack

The moduli stack of elliptic curves is a smooth separated Deligne–Mumford stack of finite type over

Spec(Z)

, but is not a scheme as elliptic curves have non-trivial automorphisms.

j-invariant

There is a proper morphism of

l{M}_1,1

to the affine line, the coarse moduli space of elliptic curves, given by the j-invariant of an elliptic curve.

Construction over the complex numbers

It is a classical observation that every elliptic curve over

is classified by its periods. Given a basis for its integral homology

\alpha,\beta\inH_1(E,Z)

and a global holomorphic differential form

\omega\in

	1
\Gamma(E,\Omega
	E)

(which exists since it is smooth and the dimension of the space of such differentials is equal to the genus, 1), the integrals

\begin\int_\alpha \omega & \int_\beta\omega \end = \begin\omega_1 & \omega_2 \end

give the generators for a

-lattice of rank 2 inside of

^[1] ^{pg 158}. Conversely, given an integral lattice

of rank

inside of

, there is an embedding of the complex torus

E_Λ=C/Λ

into

P²

from the Weierstrass P function ^{pg 165}. This isomorphic correspondence

\phi:C/Λ\toE(C)

is given by

z \mapsto [\wp(z,\Lambda),\wp'(z,\Lambda),1] \in \mathbb^2(\mathbb)

and holds up to homothety of the lattice

, which is the equivalence relation

z\Lambda \sim \Lambda ~\text~ z \in \mathbb \setminus\

It is standard to then write the lattice in the form

Z ⊕ Z ⋅ \tau

for

\tau\inak{h}

, an element of the upper half-plane, since the lattice

could be multiplied by

	-1
\omega
	1

, and

\tau,-\tau

both generate the same sublattice. Then, the upper half-plane gives a parameter space of all elliptic curves over

. There is an additional equivalence of curves given by the action of the

\text_2(\mathbb)= \left\

where an elliptic curve defined by the lattice

Z ⊕ Z ⋅ \tau

is isomorphic to curves defined by the lattice

Z ⊕ Z ⋅ \tau'

given by the modular action

\begin\begina & b \\c & d\end \cdot \tau &= \frac \\&= \tau'\end

Then, the moduli stack of elliptic curves over

is given by the stack quotient

\mathcal_ \cong[\text{SL}_2(\mathbb{Z})\backslash\mathfrak{h}]

Note some authors construct this moduli space by instead using the action of the Modular group

PSL_2(Z)=SL_2(Z)/\{\pmI\}

. In this case, the points in

l{M}_1,1

having only trivial stabilizers are dense.

Stacky/Orbifold points

Generically, the points in

l{M}_1,1

are isomorphic to the classifying stack

B(Z/2)

since every elliptic curve corresponds to a double cover of

P¹

, so the

Z/2

-action on the point corresponds to the involution of these two branches of the covering. There are a few special points^[2] ^{pg 10-11} corresponding to elliptic curves with

-invariant equal to

1728

and

where the automorphism groups are of order 4, 6, respectively^[3] ^{pg 170}. One point in the Fundamental domain with stabilizer of order

corresponds to

\tau=i

, and the points corresponding to the stabilizer of order

correspond to

\tau=e^2\pi,e^\pi

^[4] ^{pg 78}.

Representing involutions of plane curves

Given a plane curve by its Weierstrass equation $y^2 = x^3 + ax + b$ and a solution

(t,s)

, generically for j-invariant

j ≠ 0,1728

, there is the

Z/2

-involution sending

(t,s)\mapsto(t,-s)

. In the special case of a curve with complex multiplication

y^2 = x^3 + ax

there the

Z/4

-involution sending

(t,s)\mapsto(-t,\sqrt{-1} ⋅ s)

. The other special case is when

a=0

, so a curve of the form

y^2 = x^3 + b

there is the

Z/6

-involution sending

(t,s)\mapsto(\zeta₃t,-s)

where

\zeta₃

is the third root of unity

e^2\pi

Fundamental domain and visualization

There is a subset of the upper-half plane called the Fundamental domain which contains every isomorphism class of elliptic curves. It is the subset $D = \$ It is useful to consider this space because it helps visualize the stack

l{M}_1,1

. From the quotient map

\mathfrak \to \text_2(\mathbb)\backslash \mathfrak

the image of

is surjective and its interior is injective^{pg 78}. Also, the points on the boundary can be identified with their mirror image under the involution sending

Re(z)\mapsto-Re(z)

, so

l{M}_1,1

can be visualized as the projective curve

P¹

with a point removed at infinity^[5] ^{pg 52}.

Line bundles and modular functions

There are line bundles

l{L}^⊗

over the moduli stack

l{M}_1,1

whose sections correspond to modular functions

on the upper-half plane

ak{h}

. On

C x ak{h}

there are

SL_2(Z)

-actions compatible with the action on

ak{h}

given by

\text_2(\mathbb) \times \to

The degree

action is given by

\begina & b \\c & d\end : (z,\tau) \mapsto \left((c\tau + d)^kz, \frac \right)

hence the trivial line bundle

C x ak{h}\toak{h}

with the degree

action descends to a unique line bundle denoted

l{L}^⊗

. Notice the action on the factor

is a representation of

SL_2(Z)

hence such representations can be tensored together, showing

l{L}^⊗ ⊗ l{L}^⊗\congl{L}^⊗

. The sections of

l{L}^⊗

are then functions sections

f\in\Gamma(C x ak{h})

compatible with the action of

SL_2(Z)

, or equivalently, functions

f:ak{h}\toC

such that

f\left(\begina & b \\c & d \end \cdot \tau\right) = (c\tau + d)^kf(\tau)

This is exactly the condition for a holomorphic function to be modular.

Modular forms

The modular forms are the modular functions which can be extended to the compactification $\overline \to \overline_$ this is because in order to compactify the stack

l{M}_1,1

, a point at infinity must be added, which is done through a gluing process by gluing the

-disk (where a modular function has its

-expansion)^{pgs 29-33}.

Universal curves

Constructing the universal curves

l{E}\tol{M}_1,1

is a two step process: (1) construct a versal curve

l{E}_ak{h

} \to \mathfrak and then (2) show this behaves well with respect to the

SL_2(Z)

-action on

ak{h}

. Combining these two actions together yields the quotient stack

[(\text{SL}_2(\mathbb{Z}) \ltimes \mathbb{Z}^2)\backslash \mathbb{C}\times\mathfrak{h}]

Versal curve

Every rank 2

-lattice in

induces a canonical

Z²

-action on

. As before, since every lattice is homothetic to a lattice of the form

(1,\tau)

then the action

(m,n)

sends a point

z\inC

(m,n)\cdot z \mapsto z + m\cdot 1 + n\cdot\tau

Because the

\tau

ak{h}

can vary in this action, there is an induced

Z²

-action on

C x ak{h}

(m,n)\cdot (z, \tau) \mapsto (z + m\cdot 1 + n\cdot\tau, \tau)

giving the quotient space

\mathcal_\mathfrak \to \mathfrak

by projecting onto

ak{h}

SL₂-action on Z²

There is a

SL_2(Z)

-action on

Z²

which is compatible with the action on

ak{h}

, meaning given a point

z\inak{h}

and a

g\inSL_2(Z)

, the new lattice

g ⋅ z

and an induced action from

Z² ⋅ g

, which behaves as expected. This action is given by

\begina & b \\c & d\end : (m, n) \mapsto (m,n)\cdot \begina & b \\c & d\end

which is matrix multiplication on the right, so

(m,n)\cdot \begina & b \\c & d\end = (am + cn, bm + dn)

Notes and References

Book: Silverman, Joseph H. . The arithmetic of elliptic curves . 2009 . Springer-Verlag . 978-0-387-09494-6 . 2nd . New York . 405546184.
Hain. Richard. 2014-03-25. Lectures on Moduli Spaces of Elliptic Curves. math.AG. 0812.1803.
Book: Galbraith, Steven . Elliptic Curves . https://www.math.auckland.ac.nz/~sgal018/crypto-book/ch9.pdf . Mathematics of Public Key Cryptography . Cambridge University Press . The University of Auckland.
Book: Serre, Jean-Pierre . A Course in Arithmetic . 1973 . Springer New York . 978-1-4684-9884-4 . New York . 853266550.
Book: Henriques, André G . The Moduli stack of elliptic curves . Topological modular forms . Douglas, Christopher L. . Francis, John . Henriques, André G . Hill, Michael A. . 978-1-4704-1884-7. Providence, Rhode Island . 884782304. https://web.archive.org/web/20200609190825/https://www.math.ucla.edu/~mikehill/Research/surv-douglas2-201.pdf. 9 June 2020 . University of California, Los Angeles.

Moduli stack of elliptic curves explained

Properties

Smooth Deligne-Mumford stack

j-invariant

Construction over the complex numbers

Stacky/Orbifold points

Representing involutions of plane curves

Fundamental domain and visualization

Line bundles and modular functions

Modular forms

Universal curves

Versal curve

SL2-action on Z2

See also

Notes and References

SL₂-action on Z²