Elliptic Curves over C - Elliptic Curves: Number Theory and Cryptography - page 268

The theorem then says that

℘ ( z ) 2 =4 ℘ ( z ) 3 − g 2 ℘ ( z ) − g 3 . (9.5)

Therefore, the points ( ℘ ( z ) ,℘ ( z )) lie on the curve

y 2 =4 x 3 − g 2 x − g 3 .

It is traditional to leave the 4 as the coecient of x 3 , rather than performing a
change of variables to make the coe cient of x 3 equal to 1. The discriminant
of the cubic polynomial is 16( g 2 − 27 g 3 ).

PROPOSITION 9.9
Δ= g 2 − 27 g 3 =0 .

Since ℘ ( z ) is doubly periodic, ℘ ( ω i / 2) = ℘ ( −ω i / 2).

PROOF

Since

℘ ( −z )= −℘ ( z ), it follows that

℘ ( ω i / 2) = 0 ,

i =1 , 2 , 3 . (9.6)

Therefore, each ℘ ( ω i / 2) is a root of 4 x 3 g 3 , by (9.5). If we can show

that these roots are distinct, then the cubic polynomial has three distinct
roots, which means that its discriminant is nonzero. Let

− g 2 x −

h i ( z )= ℘ ( z ) − ℘ ( ω i / 2) .

Then h i ( ω i / 2) = 0 = h i ( ω i / 2), so h i vanishes to order at least 2 at ω i / 2. Since
h i ( z ) has only one pole in F , namely the double pole at z = 0, Theorem 9.1(5)
implies that ω i / 2 is the only zero of h i ( z ). In particular,

h i ( ω j / 2) =0 , when j = i.

Therefore, the values ℘ ( ω i / 2) are distinct.

The proposition implies that

E : y 2 =4 x 3

− g 2 x − g 3

is the equation of an elliptic curve, so we have a map from z C to the

points with complex coordinates ( ℘ ( z ) ,℘ ( z )) on an elliptic curve. Since ℘ ( z )
and ℘ ( z ) depend only on z mod L (that is, if we change z by an element of
L , the values of the functions do not change), we have a function from C /L
to E ( C ). The group C /L is a group, with the group law being addition of
complex numbers mod L . In concrete terms, we can regard elements of C /L
as elements of F . When we add two points, we move the result back into F by
subtracting a suitable element of L . For example, ( . 7 ω 1 + . 8 ω 2 )+( . 4 ω 1 + . 9 ω 2 )
yields . 1 ω 1 + . 7 ω 2 .

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