104 CHAPTER 9

group x ◦ y = y ◦ x for all elements x and y in the group. Because an

elliptic curve is the set of solutions to a Z-equation, we can study

how the Galois group permutes these solutions. And because it is an

abelian group, we can apply the theorems of algebra about abelian

groups. By the way, the name “elliptic curve” is derived from the

fact that certain elliptic curves are connected to the problem of

studying the arc-length of certain ellipses. An elliptic curve itself

is not elliptical in shape.

To begin, an elliptic curve can be thought of as the set of solutions

to the equation

y

2

= x

3

+ Ax + B,

where A and B can be any two ﬁxed integers, as long as 2(4A

3

+

27B

2

) = 0.

2

Usually, we use the letter E to stand for an elliptic

curve.

We often use geometric terminology for the elements of E(R),

calling them “points.” And we call the whole set E(R) a “curve.” This

is because when R is the set of real numbers R, we can actually

graph E(R) and view the solutions (x, y) as actual points on the

resulting curve.

An Example

For example, E can be the elliptic curve deﬁned by the equation

y

2

= x

3

+ 1 (so here we are choosing A = 0 and B = 1). The graph of

y

2

= x

3

+ 1 in the xy-plane is shown in Figure 9.1.

We view E as a variety, so, for example,

E(Z) ={(x, y):y

2

= x

3

+ 1, x, y ∈ Z}

E(Q) ={(x, y):y

2

= x

3

+ 1, x, y ∈ Q}

E(R) ={(x, y):y

2

= x

3

+ 1, x, y ∈ R}

E(F

5

) ={(x, y):y

2

= x

3

+ 1, x, y ∈ F

5

}

2

The reason for this rather arcane restriction is that shortly, in one of our formulas, we

might implicitly be dividing by a factor of 2(4A

3

+ 27B

2

), and we need to make sure that

we do not divide by 0.

ELLIPTIC CURVES 105

Figure 9.1: y

2

= x

3

+ 1

and so on. In other words, E(Z) tells us to search for integers that

satisfy the deﬁning equation for E, while E(C) tells us to search for

complex numbers that satisfy the same equation. However, when we

plug a number system R into E to get E(R), we are tacitly assuming

that 2(4A

3

+ 27B

2

) = 0inR. So in our example we must assume

that 6 = 0inR, so for instance R = F

2

or R = F

3

are not allowed for

this E.

It is not so clear in this example what the elements of E(Z) are.

A bit of trial and error shows that if x = 0, then y =±1 works, and

if x =−1, then y = 0 also works. A bit more trial-and-error will let

you work out that if x = 2, then y =±3 works. It is even less obvious

what the elements of E(Q) are, or even if there are any elements of

E(Q) other than those in E(Z).

On the other hand, it is a little easier to ﬁnd out about E(R):

As long as we plug in a value of x that makes x

3

+ 1 bigger than

or equal to 0 (which means that we have to pick x ≥−1), then

we can take a square root and ﬁnd a y-value. For example, if

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