We pause here to describe a class of Z-varieties called
“elliptic curves.” They are groups and varieties at the
same time, which enables us to know much more about
them than we do about most varieties. In this chapter, we
only give the basic facts. In chapter 18, we will use them
to obtain some beautiful and complicated representations
of the absolute Galois group which were essential to the
proof of Fermat’s Last Theorem.
Elliptic Curves Are “Group Varieties”
Serge Lang began his book Elliptic Curves: Diophantine Analysis by
writing: “It is possible to write endlessly about elliptic curves. (This
is not a threat.)” That is because an elliptic curve is simultaneously
an example of two different concepts that we have discussed in
earlier chapters: varieties and groups. An elliptic curve E is the set
of solutions to a certain kind of Z-equation so that for any ﬁeld R,
that is, any number system in which we can divide by any nonzero
element, E(R) is a group.
Moreover, E(R)isalwaysanabelian group, which is the term
that mathematicians give to a group in which the group law is
commutative as well as associative. That is to say, in an abelian
More exactly, E(R) is a group after you extend it by adding one more element, called O.
And there is a slight condition on R that needs to be observed, as we shall see.