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20

Families of Elliptic Curves

The purpose of this chapter is to return to the concept of families of elliptic curves in
the context of scheme theory and to point out some of the many areas of mathematics,
and now also even of physics, in which families of elliptic curves play a role. The
idea of considering an elliptic curve over two related fields, for example, Q and F p
arose in chapter 5 when studying torsion in E(Q) for an elliptic curve E over Q,and
in chapter 14 elliptic curves over a local field K were considered as objects over the
ring O of integers in K and over the residue class field k of K. Both of these cases
are included in the general concept of a morphism of schemes π : E → B having
the property that the fibres of π are elliptic curves, and we refer to π as a family of
elliptic curves.
To analyze the concept further, we could start with a morphism of schemes π :
X → B where all the fibres are curves, or with an eye towards elliptic curves, all
fibres are curves of genus one. For a family of genus one curves to be a family of
elliptic curves, we must be given a zero point on each fibre with the property that it
varies algebraically over the base scheme, that is, it is a section of the morphism π.
Since each fibre has a group law with the value of this section as zero, there must
be a morphism X × B X → X inducing the group law on each fibre. Such a general
family is not of much use without some combination of conditions on π,thatis,
flat, smooth, and proper. The N´ eron models in chapter 14, §2, is a first illustration of
such families; it is smooth, but not in general proper. In other contexts we require the
family to be flat and proper, but it is not necessary for it to be smooth.
The next direction where we restrict the problem is with the base B. The classical
literature on families of curves is very extensive for one dimensional B, for example
a curve over a field k or B = Spec(R) where R is a discrete valuation ring or more
generally any Dedekind ring. This means that X is a surface over the field k in the
first case, or X is an arithmetic surface over the Dedekind ring R. Chapters 14 and
15 can be thought of as introductions to arithmetic surfaces.
We saw in the previous chapter that the classification of surfaces X over a field
k is closely related to the existence of morphisms π : E → B onto a curve B with
given fibres. The two cases of fibres either of rational curves or elliptic curves are
especially basic. These morphisms are called fibrations when π satisfies suitable con-