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176 9. Elliptic and Hypergeometric Functions

is a relation between elliptic functions for the lattice L with all poles at points of L.
The relation is an equality because the difference between the two sides is an elliptic
function without poles which is zero at 0, and, hence, the difference is zero.

(4.2) Differential Equation for ℘(z).

3
2

℘ (z, L) = 4℘(z, L) − g 2 (L)℘(z, L) − g 3 (L),

where g 2 (L) = 60G 2 (L) and g 3 (L) = 140G 3 (L). Thus the points (℘(z), ℘ (z)) ∈
2
C lie on the curve defined by the cubic equation with g 2 = g 2 (L) and g 3 = g 3 (L)
2 3
y = 4x − g 2 x − g 3 .
For a basis L = Zω 1 +Zω 2 and ω 3 = ω 1 +ω 2 , let e i = ℘(ω i /2), where i = 1, 2, 3.
Then the elliptic function ℘(z) − e i = f (z) has a zero at ω i /2 which must be of
even order by (2.5), and thus

ω i ω i
f = ℘ = 0
2 2
for i = 1, 2, 3. By comparing zeros and poles, we obtain the factorization of (4.2):
2

℘ (z) = 4 (℘(z) − e 1 )(℘(z) − e 2 )(℘(z) − e 3 ) ,
3
where e 1 , e 2 ,and e 3 are the roots of 4x − g 2 x − g 3 . Since ℘(z) takes the value e i
with multiplicity 2 and has only one pole of order 2 modulo L, we see that e i = e j
3
for i = j. We are led to the following result since the cubic 4x − g 2 x − g 3 has
distinct roots.
(4.3) Theorem. The function h : C/L → E(C), where E is the elliptic curve over
C with equation
2
2
3
wy = 4x − g 2 (L)w x − g 3 (L)w 3
and h(z mod L) = (1,℘(z), ℘ (z)) for z /∈ L, h(0 mod L)= (0, 0, 1), is an analytic

groupisomorphism.
Proof. Clearly h(z mod L) = (0, 0, 1) ∈ E(C) is the zero element of E(C) if and
3
3
3

only if z mod L = 0 ∈ C/L,and h(z mod L) = (z : z ℘(z) : z ℘ (z)) is analytic
at 0 ∈ C/L with values in the projective plane.
To see that h : C/L − 0 → E(C) − 0 is an analytic isomorphism, we consider

(x, y) ∈ E(C)−0. There are two zeros z 1 , z 2 of the function ℘(z)−x with ℘ (z 1 ) =

−℘ (z 2 ) =±y from the equation of E. Thus h(−z mod L) =−h(z mod L) since

z 1 +z 2 ∈ L by (3.5). Note further that z 1 ≡ z 2 mod L if and only if ±℘ (z 1 ) = y = 0.
Hence h is an analytic isomorphism commuting with the operation of taking inverses.
Finally, to see that h or equivalently h −1 preserves the group structure, we study
the intersection points of the line y = λx + ν with the cubic curve E as in 1, §1. The

elliptic function f (z) = ℘ (z) − (λ℘(z) + ν) has a pole of order 3 at each point of