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§4. The Differential Equation for ℘(z) 177

L, and by (2.5) there are three zeros z 1 , z 2 ,and z 3 of f (z) with z 1 + z 2 + z 3 ∈ L.
Thus under h −1 the three intersection points of y = λx + ν, which add to zero on E,
are transformed to three points z 1 + L, z 2 + L,and z 3 + L ∈ C/L which add to zero
on C/L. This proves the theorem.
Using the above considerations in proving h −1 is a group morphism and the

techniques of 1, §2, we can derive the addition formulas for ℘(z).For ℘ (z i ) =
λ℘(z i ) + ν, where i = 1, 2, 3, the equation
3 2
4x − (λx + ν) − g 2 x − g 3 = 0
2
has three roots, and the sum of the roots is related to the coefficient of x by the
relation
λ 2
= ℘(z 1 ) + ℘(z 2 ) + ℘ (z 3 ) .
4
Given z 1 , z 2 , choose z 3 =−(z 1 + z 2 ) mod L, and for ℘(z 1 ) = ℘(z 2 ),wehave

℘ (z 1 ) − ℘ (z 2 ) = λ(℘(z 1 ) − ℘(z 2 )). Since ℘ is an even function, this gives the
following first division formula. The second formula is obtained by letting z 1 and z 2
approach z in the limit so z 3 becomes −2z.
(4.4) Addition Formulas for the Weierstrass ℘-Function.
2

1 ℘ (z 1 ) − ℘ (z 2 )
(1) ℘ (z 1 + z 2 ) =−℘(z 1 ) − ℘(z 2 ) + .
4 ℘(z 1 ) − ℘(z 2 )
2

1 ℘ (z)

(2) ℘(2z) =−2℘(z) + .
4 ℘ (z)

In 3(2.3) we introduced the notion of an admissible change of variable carrying
one normal form of the equation of an elliptic curve into another or equivalently
defining an isomorphism between two elliptic curves.
(4.5) Remark. An admissible change of variable defining an isomorphism f :
2
2
3
3
E → E, where E is given by y = 4x −g 2 x −g 3 and E by y = 4x −g x −g ,

2 3
3
2
has the form xf = u x and yf = u y . Moreover, we have by 2(2.4) the relations
g 2 = u g and g 3 = u g .
4
6
2 3

In (1.3) we considered the equivalence between two lattices L and L together
with the corresponding isomorphism defined between their related tori T = C/L

and T = C/L . Under this isomorphism there are transformation relations for the
Weierstrass ℘-function and its related coefficient functions.
(4.6) Proposition. For two equivalent lattices L and L = λL the following rela-

tions hold:
2
3
℘(z, L) = λ ℘(λz,λL), ℘ (z, L) = λ ℘ (λz,λL),

4
6
2k
G 2k (L) = λ G 2k (λL), g 2 (L) = λ g 2 (λL), and g 3 (L) = λ g 3 (λL).