Elliptic Curves over C - Elliptic Curves: Number Theory and Cryptography - page 264

problem if w lies on a side of F . Suppose w = xω 1 with 0 <x< 1. Then
ω 3 − w =(1 − x ) ω 1 + ω 2 ∈ F . In this case, we translate by ω 2 to get

(1 x ) ω 1 ,and

again the two elements of the pair are distinct. The case w = xω 2 is handled
similarly.
For a fixed w , the function ℘ ( z ) − ℘ ( w ) has zeros at z = w and z = ω 3 − w .
By Theorem 9.1(5), these are the only two zeros in F and they are simple
zeros. Therefore, the function
h ( z )=
( w, ω 3 −

− x ) ω 1 ∈ F .Since w = ω 1 / 2, we have x =1 / 2, hence xω 1 =(1 −

℘ ( w )) ord w f ( ℘ ( z ) −

w )

(the product is over pairs ( w, ω 3 − w )) has a zero of order ord w f at w and
at ω 3 − w when ord w f> 0andhasapoleofthesameorderas f when

ord w f< 0. Since ord w f = 0 by Theorem 9.1, the poles at z ∈ L of the
factors in the product cancel. Therefore, f ( z ) /h ( z ) has no zeros or poles in F .
By Theorem 9.1(1), f ( z ) /h ( z ) is constant. Since h ( z ) is a rational function
of ℘ ( z ), so is f ( z ). This completes the proof of Theorem 9.3.

In order to construct functions with prescribed properties, it is convenient
to introduce the Weierstrass σ -function. It is not doubly periodic, but it
satisfies a simple transformation law for translation by elements of L .

PROPOSITION 9.5
Let 1

ω e ( z/w )+ 2 ( z/w ) 2 .

σ ( z )= σ ( z ; L )= z
ω∈L
ω

z

−

=0

Then

1. σ ( z ) isanalyticfor all z ∈ C

2. σ ( z ) has simp e zeros at each ω ∈ L and has no other zeros

d 2

dz 2 log σ ( z )=

− ℘ ( z )

3.

4. given ω

∈ L ,there exist a = a ω and b = b ω su ch that

σ ( z + ω )= e az + b σ ( z )

for all z ∈ C .

PROOF The exponential factor is included to make the product converge.
A short calculation yields the power series expansion

(1 − u ) e u + 2 u 2 =1+ c 3 u 3 + c 4 u 4 + ··· .