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140 6. Proof of Mordell’s Finite Generation Theorem

h E (P + Q) + h E (P − Q) = h(θ(P + Q, P − Q)) = h(θ(u(P, Q)))
= h( f (θ(P, Q))) ∼ 2h(θ(p, Q)).

The last expression can be studied using (6.5), and we have
2h(θ(P, Q)) = 2h(s(q(P), q(Q))) ∼ 2h(q(P)) + 2h(q(Q))
= 2h E (P) + 2h E (Q).

Now we apply (5.5) to see that h E is quasiquadratic and further (5.6) to see that h E
is a quadratic function.
Finally, the function h E is unique since two possible functions would be quadratic
by the above argument and equivalent to h , and hence they are equivalent to each

other. If two quadratic functions differ by a bounded function, then they are equal.
This proves the theorem.
(7.3) Corollary. With the hypothesis and notations of the previous theorem, the
√
function |P|= h E (P) is a norm on E(k).
Now we assume Theorems (3.2) and (7.2) to deduce the following theorem of
Mordell. This theorem was generalized to number fields by A. Weil and is one of the
main results of this book.

(7.4) Theorem (Mordell–Weil). Let E be an elliptic curve over a number field k.
Then the groupE(k) is finitely generated.

Proof of (7.4) for k = Q where E = E[a, b]. By (3.2) the index (E(Q) :2E(Q))
√
is finite. By (7.2) and (7.3) the function |P|= h E (P) is a norm on E(Q).The
criterion (1.3) applies to show that (E(Q) is a finitely generated abeian group. This
proves the theorem in this case.

Sketch of proof for k any number field and any E over k. In the general case we
2

extend k to k so that the equation of E has the form y = (x −α)(x −β)(x −γ).By
√

(4.7) the index E(k ) :2E(k )) is finite. Again by (7.3) the function |P|= h E (P)
is a norm on E(k ); here we assume there is a height function on projective space

over k, and (6.5) holds in order to carry out the proof of (7.2) for a number field.

Finally, E(k) is then a subgroup of a finitely generated group E(k ), and thus, it is
finitely generated.
§8. The Canonical Height on Projective Spaces over Global
Fields

In order to formulate the notion of height in the case of number fields and more gen-
erally global fields, we reformulate the definition over the rational numbers without
using Z-reduced coordinates of a point but using instead all the valuations or absolute
values on Q. Up to equivalence, each absolute value on Q is either of the form: