Elliptic Curves over C - Elliptic Curves: Number Theory and Cryptography - page 296

PROOF As in the proof of Lemma 9.28, we see that

f 2 n +1
f n +1 f n
since the two sides have the same divisors and their expansions at 0 have the
same leading coecient. Since

℘ (( n +1) z ) − ℘ ( nz )= −

℘ (( n +1) z ) − ℘ ( nz )=( ℘ (( n +1) z ) − ℘ ( z )) − ( ℘ ( nz ) − ℘ ( z ))

f n +2 f n
f n +1

+ f n +1 f n − 1
f n

= − ,

the result follows by equating the two expressions for ℘ (( n +1) z ) − ℘ ( nz ).

LEMMA 9.30
℘ f 2 n =( f n )( f n +2 f n− 1 − f n− 2 f n +1 ) .

PROOF As in the proofs of the previous two lemmas, we have

℘ f 2 n

f n− 1 f n +1 .
(A little care is needed to handle the points ω j / 2.) Since

℘ (( n +1) z ) − ℘ (( n − 1) z )= −

℘ (( n +1) z ) − ℘ (( n − 1) z )=( ℘ (( n +1) z ) − ℘ ( z )) − ( ℘ (( n − 1) z ) − ℘ ( z ))

f n +2 f n
f n +1

+ f n f n − 2
f n− 1

= − ,

the result follows.

LEMMA 9.31
For all n ≥ 1 ,

f n ( z )= ψ n ℘ ( z ) ,

2 ℘ ( z )

1

where ψ n is defined inSection 3.2.

PROOF Since ψ 1 =1and ψ 2 =2 y , the lemma is easily seen to be true for
n =1 , 2. From Equations (9.10) and (9.8) in Section 9.2, we have

f 3
( ℘ ) 2

f 3
f 2

− = − = ℘ (2 z ) − ℘ ( z ) (9.25)

℘ ( z )
℘ ( z )

2

1
4

− 2 ℘ ( z ) − ℘ ( z )

=

3 ℘ 4

3

2 g 2 ℘ 2 16 g 2

1

− − 3 g 3 ℘ −

= − .

( ℘ ) 2