Let be a scheme. An **elliptic curve over ** should be thought of as a continuously varying family of elliptic curves parametrized by .

Definition 1Anelliptic curve overis a proper, flat morphism whose geometric fibers are curves of genus one together with a section .

This is a reasonable notion of “family”: observe that a morphism can be used to pull back elliptic curves over . The flatness condition can be thought of as “continuity.” For an algebraically closed field, this reduces to the usual notion of an elliptic curve.

A basic property of elliptic curves over algebraically closed fields is that they imbed into and are cut out by (nonsingular) Weierstrass equations of the form

This equation is unique up to an action of a certain four-dimensional group of transformations. The first goal is to show that, locally, the same is true for an elliptic curve over a base.

**1. The zero section**

Let be an elliptic curve over the scheme , with zero section . We are going to find an ample line bundle on . To do so, observe that (as a section to a separated morphism) imbeds inside as a closed subscheme. Observe that is smooth over , since it is flat and the fibers are smooth curves. The zero section is thus cut out locally by one equation.

It follows now that can be used to define a line bundle on , which is compatible with base-change on . When restricted to each fiber (an elliptic curve), it is the line bundle associated to the divisor at the origin . There is a natural morphism .

Now consider the line bundle . When restricted to each geometric fiber , this is a very ample line bundle on the elliptic curve associated to three times the origin. By Riemann-Roch, we have

for every geometric fiber.

By the theorem of cohomology and base change, we find that \( f_* \mathcal{L}^{3} \) is a vector bundle of rank three on , and that the maps

are isomorphisms: that is, any section along extends to a section in some neighborhood. Moreover, we find that the formation of the pushforward commutes with base change .

It follows now from very ampleness on the fibers that is a surjection of sheaves, so defines a map of -schemes

Since commutes with base change, we find that is a closed imbedding when restricted to any fiber. Hence, is an imbedding and is very ample.

Thus, given an elliptic curve over , we get a *canonical* imbedding into the projectivization of a three-dimensional vector bundle on .

**2. Weierstrass equations**

Suppose now given a trivialization , given by three sections generating . Assume that , in fact (recall that there is a map ). Locally, we can always choose such a trivialization, but it is not unique. Then we have an imbedding of -schemes

given by the sections generating . Suppose is, further, in the image of (which is locally free of rank two).

The claim now is that is cut out as a subscheme of by a Weierstrass equation. Here the reasoning is analogous as in the ordinary case when . In fact, we note that

for each , by a Riemann-Roch calculation (and the ‘s are zero). It follows that is locally free of rank six. The elements are *seven* sections and consequently must be linearly dependent. They generate the vector bundle, too: this can be checked stalkwise and then one notes that the orders at zero are anywhere between and .

Locally on , we can find a relation of linear dependence (unique up to scaling) among all these involving functions of . That is, we can find a relation

Now, the claim is that are necessarily invertible. In fact, we can check this after restricting to the geometric fibers: if when restricted to one geometric fiber exactly one of vanished, then one side would have order six at the origin and one side would have order . vanished, the left side and the right side could not have the same order either.

Anyway, rescaling we can get the appropriate equation where .

Proposition 2Let be an elliptic curve over . Then, for each , there is an open neighborhood containing such that the elliptic curve has a Weierstrass equation as above.

In other words, locally on , looks like the zero locus of a Weierstrass equation when the are taken as functions on (an open subset of) .

There is thus a “Zariski locally universal” elliptic curve. Consider the scheme and the subscheme of cut out by the Weierstrass equation . We have then a proper, flat morphism

Properness is clear, and flatness follows from the fact that the fibers all have the same Hilbert polynomial (they are cubic curves in ). There is an open locus over which the fibers are smooth (i.e., where suitable discriminant polynomials are nonzero), and the restriction of is then an elliptic curve. The zero section is given by the point at infinity.

Here the line bundle satisfies restricted to , and we take the sections to imbed in .

**3. Uniqueness**

The choice of a Weierstrass equation is not unique. Given an elliptic curve , certain things were canonical: the line bundle , the three-dimensional vector bundle on , and the imbedding

The element is canonical, but the choice of is not (over a sufficiently small base that is free). Given another choice which satisfy a Weierstrass equation, we have

where are units. This follows because generate the same two-dimensional subbundle . Since are required to satisfy a Weierstrass equation, we must have

this means that we can parametrize the pair via . It follows that the general change of variables is

These transformations are parametrized by a *group scheme*, smooth and of finite type over .

**4. The stack of elliptic curves**

We define the following (pseudo)-functor

To a ring , we set to be the collection of all elliptic curves over , together with all -isomorphisms between them; the pull-back is the pull-back of schemes. The claim is that is a Deligne-Mumford stack.

First, we should see that is even a stack: given an étale morphism , we should be able to “descend” elliptic curves over to elliptic curves over .

Proposition 3is a stack for the flat topology.

*Proof:* Suppose given a faithfully flat morphism , and suppose given an elliptic curve together with an isomorphism over satisfying the cocycle condition. Let be the three-dimensional vector bundle over associated functorially to ; this comes equipped with the descent data to become canonically the base-change of an -module . It follows that we really have a descent problem for subschemes of , which we can solve using standard flat descent.

In order to see this, we will need to produce a cover of . We will just produce a smooth cover here. Consider the map

classifying the elliptic curve which was “universal” for Weierstrass elliptic curves. The claim is that this is smooth and surjective. It is surjective because any elliptic curve is locally Weierstrass: stated stackily, any map factors locally through .

To see that it is smooth, we will argue more strongly, that is the quotient stack for a suitable (smooth) group scheme over . In fact, this follows from our earlier analysis: étale (or Zariski) locally, any elliptic curve comes from a Weierstrass equation, and this Weierstrass equation is unique up to the action of , where is the group of transformations of Weierstrass equations as above.

(Actually, so far we have only proved that it is an Artin stack.)

June 13, 2012 at 9:33 pm

These are great notes. Can you give a reference?

June 14, 2012 at 6:19 am

Andre Henriques has some nice notes: http://math.mit.edu/conferences/talbot/2007/tmfproc/Chapter04/henriques.pdf. Otherwise, I only know of Deligne-Mumford and Deligne-Rapoport.

June 14, 2012 at 11:52 am

Actually, ch. 2 of Katz-Mazur’s “Arithmetic moduli of elliptic curves” is very nice (I think it was where I learned this material).