ELLIPTIC CURVES 109

The particular details of the abelian group law actually do not

concern us very much. What does matter is that if we combine two

points in E(Q), for example, then the result is also in E(Q). This

is not true for E(Z) because the number λ might be a fraction, and

then we might get fractional values for x

3

and y

3

. But it is true

for any number system in which we can divide. So if we start with

elements of E(R), and add them, we will get elements of E(R); ditto

for E(C) and E(F

p

).

Digression: What Is So Great about Elliptic Curves?

The question should be, “What is not great about them?” They were

essential in Wiles’s proof of Fermat’s Last Theorem. You can read

a brief description in our paper (Ash and Gross, 2000), or look at

some of the many books on the subject (Hellegouarch, 2002; Singh,

1997; van der Poorten, 1996), or skip ahead to chapter 22. Elliptic

curves have given rise to a tremendous amount of interesting

number theory, including the part we are trying to explain in this

book. Mathematicians are just lucky that elliptic curves are fairly

simple—they only involve two variables and no powers higher than

the cube—and yet are so rich.

One can see how people began to be curious about elliptic

curves—even ancient Greeks such as Diophantus, although he did

not call them elliptic curves. As we have seen, equations in only

one variable can be difﬁcult to solve; even formulas for the cubic

and quartic equations were not discovered until the Renaissance.

But we go on to one equation in two variables anyway. If all

the exponents are 1 or less, the solution set can be found by

simple arithmetic. Uninteresting. (See chapter 10 about how you

can use matrices to solve such equations.) If all the exponents

are 2 or less, the solution set can be understood in terms of

what are called “conic sections,” and some of the later Greeks

understood those very well, although it was a fairly advanced

topic for them. So the natural thing to do, if you are a mathe-

matician, is to keep going until you get stuck: Throw in a cubic

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