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Fearless Symmetry by Robert Gross, Avner Ash

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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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