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6

Proof of Mordell’s Finite Generation Theorem

In this chapter we prove that the rational points E(Q) on an elliptic curve E over the
rational numbers Q form a finitely generated group. We will follow a line of argu-
ment which generalizes to prove A. Weil’s extension to number fields: The Mordell–
Weil group E(k) of points over a number field k on an elliptic curve E is a finitely
generated group.
There are two conditions on an abelian group A which together are equivalent
to A being finitely generated. First the index (A : nA) is finite for some n > 1,
usually one shows that (A :2A) is finite, and second the existence of a norm on the
group. This last metric property allows a descent procedure to take place similar to
the descent arguments first introduced by Fermat.
The proof of the finiteness of (E(k) :2E(k)) is done over any number field. The
argument uses the equation of E in factored Weierstrass form
2
y = (x − α)(x − β)(x − γ).
A factored form of the equation is always possible after at most a ground field ex-
tension of degree 6. Thus we are led to number fields even if our interest is elliptic
curves over Q.
The norm is constructed from the canonical height function of projective space.
The subject of heights comes up again in considerations related to the Birch and
Swinnerton–Dyer conjectures in Chapter 17.

§1. A Condition for Finite Generation of an Abelian Group

Multiplication by a natural number m on an abelian group A is a homomorphism
m
A → A with a kernel m A and a cokernel A/mA. The order of A/mA is also the
index (A : mA) of the subgroup mA in A.If A is finitely generated, then (A : mA)
is finite for all nonzero m.

(1.1) Definition. A norm function | | on an abelian group A is a function | | :
A → R such that: