Elliptic Curves over C - Elliptic Curves: Number Theory and Cryptography - page 297

Therefore,

3 1

f 3 =3 ℘ 4

16 g 2
=3 ℘ 4 +6 A℘ 2 +12 B℘ − A 2 = ψ 3 ( ℘ ) .

2 g 2 ℘ 2

− − 3 g 3 ℘ −

This proves the lemma for n =3.
By Equation (9.7) in Section 9.2, we have

℘ (2 z ) 2

℘ ( z )

℘ (2 z + z )= 1
4

−

− ℘ (2 z ) − ℘ ( z )

℘ (2 z ) − ℘ ( z )

and

℘ (2 z )+ ℘ ( z )
℘ (2 z ) − ℘ ( z )

2

℘ (2 z − z )= 1
4

− ℘ (2 z ) − ℘ ( z ) .

Therefore,

f 4 f 2
f 3

− = ℘ (3 z ) − ℘ ( z )

℘ (2 z ) − ℘ ( z )
℘ (2 z ) − ℘ ( z )

2 ℘ (2 z )+ ℘ ( z )

℘ (2 z ) − ℘ ( z )

2

1
4

1
4

= −

℘ (2 z ) ℘ ( z )
( ℘ (2 z ) − ℘ ( z )) 2

= −

℘ (2 z ) ℘ ( z )
( −f 3 /℘ ( z ) 2 ) 2

= − (by Equation 9.25)

℘ (2 z ) ℘ ( z ) 5
f 3

= − .

1

This yields f 4 f 2 = ℘ (2 z ) ℘ ( z ) 5 . We know that

2 ℘ (2 z )isthe y -coordinate of

2 ℘ ( z )), which means that ℘ (2 z ) can be expressed in terms of ℘ ( z )
and ℘ ( z ), using the formulas for the group law. When this is done, we obtain
f 4 = ψ 4 ℘, 1

1

2( ℘ ( z ) ,

2 ℘ ,

so the lemma is true for n =4.
Since the f n 's satisfy the same recurrence relations as the ψ n 's (see Lem-
mas 9.29 and 9.30 and the definition of the ψ n 's), and since the lemma holds
for enough small values of n , the lemma now follows for all n .

LEMMA 9.32

℘ ( nz )= f 2 n
f n

.