Moduli stack of elliptic curves

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

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

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

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

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

and a global holomorphic differential form

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

It is standard to then write the lattice in the form

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

Then, the upper half-plane gives a parameter space of all elliptic curves over

Then, the moduli stack of elliptic curves over

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

since every elliptic curve corresponds to a double cover of

-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

where the automorphism groups are of order 4, 6, respectively[3] pg 170.

One point in the Fundamental domain with stabilizer of order

, and the points corresponding to the stabilizer of order

In the special case of a curve with complex multiplication

There is a subset of the upper-half plane called the Fundamental domain which contains every isomorphism class of elliptic curves.

It is useful to consider this space because it helps visualize the stack

is surjective and its interior is injective[4]pg 78.

Also, the points on the boundary can be identified with their mirror image under the involution sending

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

whose sections correspond to modular functions

action descends to a unique line bundle denoted

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

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

Combining these two actions together yields the quotient stack

Fundamental domains of the action of on the upper half-plane are shown here as pairs of ideal triangles of different colors sharing an edge. The "standard" fundamental domain is shown with darker edges. Suitably identifying points on the boundary of this region, we obtain the coarse moduli space of elliptic curves. The stacky points at and are on the boundary of this region.