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126 6. Proof of Mordell’s Finite Generation Theorem

(1) |P|≥ 0 for all P ∈ A, and for each real number r the set of P ∈ A with
|P|≤ r is finite.
(2) |mP|=|m||P| for all P ∈ A and integers m.
(3) |P + Q|≤|P|+|Q| for all P, Q ∈ A.
Observethatfromthese axioms P is a torsion point if and only if |P|= 0. By
the first axiom the torsion subgroup Tors(A) of A is therefore finite.
n
n
n
(1.2) Examples. The restriction of the Euclidean norm on R to Z is a norm on Z .
The zero norm is a norm on any finite abelian group, and, in fact, it is the only norm.
If A i is an abelian group with norm || i for i = 1,..., n, then A 1 ⊕· · · ⊕ A n = A
has a norm given by |(P 1 ,..., P n )|=|P 1 | 1 +· · · +|P n | n . In particular, any finitely
n
generated abelian group A has a norm since A is isomorphic to Z × Tors(A).The
converse is the next proposition.
(1.3) Proposition. An abelian groupA is finitely generated if and only if the index
(A : mA) is finite for some m > 1 and the groupA has a norm function.
Proof. The direct implication is contained in the above discussion. Conversely,
let R 1 ,..., R n be representatives in A for the cosets modulo mA, and let c =
max i |R i |+ 1. Let X denote the finite set of all P ∈ A with |P|≤ c, and let G
be the subgroup of A generated by X. In particular, each R i ∈ X ⊂ G.
If there exists P ∈ A − G, then there exists such a P with minimal norm by the
first axiom. Since P ≡ R i (mod mA) for some coset representative of A/mA,we
have P = R i + mQ or mQ = P + (−R i ),and
m|Q|=|mQ|≤|P|+|R i | < |P|+ c ≤ m|P|.
From the minimal character of P ∈ A − G, we see that Q ∈ G, and hence the
sum P = R i + mQ is also in G. This contradicts the assumption that A − G is
nonempty and, therefore, A = G, because G is finitely generated by X. This proves
the proposition.
(1.4) Remark. The above argument has a constructive character and is frequently
referred to as a descent procedure. For a given P ∈ A we can choose a sequence

P = R i(0) + mP 1 , P 1 = R i(1) + mP 2 ,..., P j = R i( j) + mP j+1 ,...
with
|P| > |P 1 | > |P 2 | > ··· > |P j | > |P j+1 | > ···
and the descent stops when some |P j |≤ c. Thus P is a sum of chosen coset repre-
sentatives.
In the next section we show how descent was first used by Fermat to show the
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nonexistence of solutions to x + y = 1, where x and y are rational numbers both
nonzero. The program for the remainder of this chapter is to check the two conditions
of (1.3) for A = E(k) the rational points on an elliptic curve over a number field.
r
(1.5) Remark. If the index (A : pA) = p for a prime number p, then r is an upper
bound for the rank of A. This principle is used frequently in obtaining information
on the rank of the group of rational points on an elliptic curve especially for p = 2.