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130 6. Proof of Mordell’s Finite Generation Theorem

2
2 ord p (y) = ord p (y ) = ord p (x − r i ),
and thus all ord p (x − r j ) are even. This proves the proposition.
(4.2) Notations. Let E be an elliptic curve defined by the equation
2
y = (x − r i )(x − r 2 )(x − r 3 )
where each r i ∈ R.
(a) Let P(E) denote the set of all irreducibles p (defined up to units in R) such
that p divides some r i −r j , where i = j. Then P(E) is a finite set. Let A(E) denote
∗ 2
∗ 2
∗
the subgroup of all cosets a(k ) in k /(k ) such that ord p (a) is even for p /∈ P(E).
∗ 2
∗
(b) Let θ 1 ,θ 2 ,θ 3 : E(k) → A(E) ⊂ k /(k ) be three functions given by the
relations:
(1) θ i (0) = 1;
∗ 2
(2) θ i ((r i , 0)) = (r j − r i )(r k − r i ) mod (k ) for {i, j, k}={1, 2, 3};
∗ 2
(3) θ i ((x, y)) = (x − r i ) mod (k ) otherwise.
Observe that the set P(E) is close to the set of irreducibles P where the curve E
has bad reduction and where k(p) has characteristic 2.
(4.3) Proposition. With the notations of (4.2) the functions θ i : E(k) → A(E) are
grouphomomorphisms and

ker(θ 1 ) ∩ ker(θ 2 ) ∩ ker(θ 3 ) ⊂ 2E(k).
Proof. Consider three points P i = (x i , y i ) on E(k) ∩ L, where L is a line and show
∗ ∗ 2
that θ i (P 1 ), θ i (P 2 ), θ i (P 3 ) ∈ k /(k ) . The line is vertical if and only if some P j =
∗ 2
∗
0, and then by inspection θ i (P 1 )θ i (P 2 )θ i (P 3 ) = 1in k /(k ) . Otherwise the line is
2
of the form y = λx +β,and x 1 , x 2 ,and x 3 are the roots of the equation (λx +β) =
f (x), where f (x) = (x − r 1 )(x − r 2 )(x − r 3 ). Hence x 1 − r i , x 2 − r i , x 3 − r i are
roots of the equation
2
3
2
(λ(x + r i ) + β) = f (x + r i ) = x + ax + bx,
where f (r i ) = 0. Collecting terms, we obtain
3
2
2
2
0 = x + (a − λ )x + (b − 2λ(λr i + β))x − (λr i + β) ,
and this leads to the following cases.
Case 1. All P j = (r j , 0) for j = 1, 2, 3. Then we calculate
2
θ i (P 1 )θ i (P 2 )θ i (P 3 ) = (x 1 − r i )(x 2 − r i )(x 3 − r i ) =−[−(r i + β) ]
∗ 2
≡ 1mod(k ) .
Case 2. Some P j = (r i , 0), which we can take to be P i = (r i , 0).Then0, x 2 −
r 1 , x 3 − r 1 are the roots of the cubic which means β =−λr i , and the equation
3
2
2
becomes 0 = x + (a − λ )x + bx.Now we have