Elliptic Curves over Q - Elliptic Curves: Number Theory and Cryptography - page 204

E 1 has order n .Since n is not a power of p ,we
may multiply P by the largest power of p dividing n and obtain a point, not
equal to

PROOF Suppose P ∈

∞ ,oforderprimeto p . Therefore, we may assume that P has order

n with p n .Let r be the largest integer such that P ∈ E r .Then

nλ r ( P )= λ r ( nP )= λ r ( ∞ ) ≡ 0(mod p 4 r ) .

0(mod p 4 r ), so P Since p E 5 r .Since5 r>r ,this

contradicts the choice of r . Therefore, P does not exist.

n ,wehave λ r ( P ) ≡ ∈

The following theorem was proved independently by Lutz and Nagell in the
1930s. Quite often it allows a quick determination of the torsion points on an
elliptic curve over Q . See Section 9.6 for another method.

THEOREM 8.7 (Lutz-Nagell)
Let E be given by y 2 = x 3 + Ax + B with A, B ∈ Z .Let P =( x, y ) ∈ E ( Q ) .
Suppose P has finiteorder.Then x, y ∈ Z .If y =0 then

y 2 4 A 3 +27 B 2 .

|

PROOF Suppose x or y is not in Z . Then there is some prime p dividing
the denominator of one of them. By part (2) of Theorem 8.1, P ∈ E r for
some r 1. Let be a prime dividing the order n of P .Then Q =( n/ ) P

has order . By Corollary 8.6, = p . Choose j such that Q

≥

∈ E j ,Q ∈ E j +1 .

Then λ j ( Q ) ≡ 0(mod p ), and

pλ j ( Q )= λ j ( pQ ) ≡ 0(mod p 4 j ) .

Therefore,

0(mod p 4 j− 1 ) . λ j ( Q ) ≡

This contradicts the fact that λ j ( Q ) ≡ 0(mod p ). It follows that x, y ∈ Z .
Assume y =0. Then2 P =( x 2 ,y 2 ) = ∞ .Since2 P has finite order,

x 2 ,y 2 ∈ Z . By Theorem 3.6,

x 2 = x 4

2 Ax 2 8 Bx + A 2 − −

.

4 y 2

Since x 2 ∈ Z , this implies that

y 2 |x 4 − 2 Ax 2 − 8 Bx + A 2 .

A straightforward calculation shows that

(3 x 2 +4 A )( x 4 2 Ax 2 8 Bx + A 2 ) (3 x 3 27 B )( x 3 + Ax + B )

=4 A 3 +27 B 2 .

− − − − 5 Ax −