Moduli stack of elliptic curves


In mathematics, the moduli stack of elliptic curves, denoted as or, is an algebraic stack over classifying elliptic curves. Note that it is a special case of the Moduli stack of algebraic curves. 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.

Properties

Smooth Deligne-Mumford stack

The moduli stack of elliptic curves is a smooth separated Deligne–Mumford stack of finite type over, but is not a scheme as elliptic curves have non-trivial automorphisms.

j-invariant

There is a proper morphism of 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 and a global holomorphic differential form , the integrals
give the generators for a -lattice of rank 2 inside of pg 158. Conversely, given an integral lattice of rank inside of, there is an embedding of the complex torus into from the Weierstrass P function pg 165. This isomorphic correspondence is given by
and holds up to homothety of the lattice, which is the equivalence relation
for
It is standard to then write the lattice in the form for, an element of the upper half-plane, since the lattice could be multiplied by, and 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
where an elliptic curve defined by the lattice is isomorphic to curves defined by the lattice given by the modular action
Then, the moduli stack of elliptic curves over is given by the stack quotient
Note some authors construct this moduli space by instead using the action of the Modular group. In this case, the points in having only trivial stabilizers are dense.

Stacky/Orbifold points

Generically, the points in are isomorphic to the classifying stack since every elliptic curve corresponds to a double cover of, so the -action on the point corresponds to the involution of these two branches of the covering. There are a few special points pg 10-11 corresponding to elliptic curves with -invariant equal to and where the automorphism groups are of order 4, 6, respectively pg 170. One point in the Fundamental domain with stabilizer of order corresponds to, and the points corresponding to the stabilizer of order correspond to pg 78.

Representing involutions of plane curves

Given a plane curve by its Weierstrass equation
and a solution, generically for j-invariant, there is the -involution sending. In the special case of a curve with complex multiplication
there the -involution sending. The other special case is when, so a curve of the form
there is the -involution sending where is the third root of unity.

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
It is useful to consider this space because it helps visualize the stack. From the quotient map
the image of is surjective and its interior is injectivepg 78. Also, the points on the boundary can be identified with their mirror image under the involution sending, so can be visualized as the projective curve with a point removed at infinitypg 52.

Line bundles and modular functions

There are line bundles over the moduli stack whose sections correspond to modular functions on the upper-half plane. On there are -actions compatible with the action on given by
The degree action is given by
hence the trivial line bundle with the degree action descends to a unique line bundle denoted. Notice the action on the factor is a representation of on hence such representations can be tensored together, showing. The sections of are then functions sections compatible with the action of, or equivalently, functions such that
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
this is because in order to compactify the stack, a point at infinity must be added, which is done through a gluing process by gluing the -disk pgs 29-33.

Universal curves

Constructing the universal curves is a two step process: construct a versal curve and then show this behaves well with respect to the -action on. Combining these two actions together yields the quotient stack

Versal curve

Every rank 2 -lattice in induces a canonical -action on. As before, since every lattice is homothetic to a lattice of the form then the action sends a point to
Because the in can vary in this action, there is an induced -action on
giving the quotient space
by projecting onto.

SL2-action on Z2

There is a -action on which is compatible with the action on, meaning given a point and a, the new lattice and an induced action from, which behaves as expected. This action is given by
which is matrix multiplication on the right, so