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§4. Finiteness of the Index (E(k) :2E(k)) 131

(x 2 − r 1 )(x 3 − r 1 ) = b = (r 2 − r 1 )(r 3 − r 1 ),

and, using this, we calculate for {i, j, k}={1, 2, 3}

θ i (P i )θ i (P j )θ i (P k ) = (r j − r i )(r k − r i )(x j − r i )(x k − r i )
2
= (r j − r i ) (r k − r i ) 2
∗ 2
≡ 1mod(k ) .

Hence each θ i is a group morphism.
The last statement follows from 1(4.1), and this proves the proposition.

(4.4) Remark. The three morphisms of the previous proposition collect to define a
group homomorphism

3
θ = (θ 1 ,θ 2 ,θ 3 ) : E(k) → A(E) ,
where ker(θ) ⊂ 2E(k) by the previous proposition. Thus E(k)/2E(k) is a subquo-
3
tient of A(E) , and (E(k) :2E(k)) is finite whenever A(E) is finite.
(4.5) Remark. The group A(E) is finite for any principal ring R where each k(p)
∗ 2
∗
and R /(R ) are finite. For example, if R = Z and k = Q, then the cardinality
of A(E) is 2 m+1 , where m is the number of primes in P(E). Hence we have the
following assertion which we will generalize immediately.
2
(4.6) Assertion. Let y = (x − r 1 )(x − r 2 )(x − r 3 ) define an elliptic curve E over
Q where each r i ∈ Z. Then the index (E(Q) :2E(Q)) is finite.
2
To prove this assertion for y = f (x) where the cubic f (x) does not necessarily
factor over Q, we will extend the ground field to k a number field, and factor the
cubic in this field k. An extension of degree 6 will be sufficient. Thus the following
more general result would be necessary even if our primary interest is elliptic curves
over Q.

(4.7) Theorem. Let E be an elliptic curve over an algebraic number field k. Then
the index (E(k) :2E(k)) is finite.
2
Proof. We can assume that E is defined by an equation y = f (x), where f (x) is a
cubic with three integral roots in k.Wetakefor R in k the principal ideal ring equal
to the ring of integers in k with a finite set of primes in k localized. By the finiteness
of the ideal class group we could localize those primes which divide a finite set of
representatives of the ideal class group. With a zero ideal class group the ring is
principal.
The group of units R is finitely generated by the Dirichlet unit theorem, and
∗
∗ 2
thus, R /(R ) is finite. Now A(E) is finite by (4.6) and we can apply (4.4) to obtain
∗
the proof of the theorem.