28 2. Elliptic curves

Q. Let R be the third point of intersection of L and E. Then R is

also a rational point in E(Q). Let L be the line through R and O.

We define P + Q to be the third point of intersection of L and E.

Notice that any vertical line x = a in the aﬃne plane passes through

[0,1,0], because the same line in projective coordinates is given by

x = az and [0,1,0] belongs to such line. Thus, if E is given by a

model

y2

=

x3

+ Ax + B, and O is chosen to be the point [0,1,0],

then L is always a vertical line, so P + Q is always the reflection of

R with respect to the x axis.

The next step in the study of the structure of E(Q) was conjec-

tured by Jules Poincaré in 1908, proved by Louis Mordell in 1922 and

generalized by André Weil in his thesis in 1928:

Theorem 2.4.3 (Mordell-Weil). E(Q) is a finitely generated abelian

group. In other words, there are points P1,...,Pn such that any other

point Q in E(Q) can be expressed as a linear combination

Q = a1P1 + a2P2 + · · · + anPn

for some ai ∈ Z.

The group E(Q) is usually called the Mordell-Weil group of E,

in honor of the two mathematicians who proved the theorem.

Example 2.4.4. Consider the elliptic curve E/Q given by the Weier-

strass equation

y2

+ y =

x3

− 7x + 6.

The set of rational points E(Q) for this elliptic curve is infinite. For

instance, the following points are on the curve:

(1,0), (2,0), (0, −3), (−3, −1), (8, −22), (−2, −4), (3, −4),

(3,3), (−1, −4), (1, −1), (0,2), (2, −1), (−2,3), (−1,3),

1

4

,

13

8

,

25

9

, −

91

27

, −

26

9

,

28

27

,

7

9

,

17

27

, . . . .

At a first glance, it may seem very diﬃcult to describe all the points

on E(Q), including those listed above, in a succinct manner. However,

the Mordell-Weil theorem tells us that there must be a finite set of

points that generate the whole group. Indeed, it can be proved that