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178 9. Elliptic and Hypergeometric Functions

Proof. These formulas follow easily from the definitions. For example, calculate

1 1
−3

℘ (λz,λL) =−2 = λ (−2)
(λz − λω) 3 (z − ω) 3
λω∈λL−0 ω∈L−0
−3

= λ ℘ (z, L).
Now we can make precise how the isomorphism h in (4.3) relates to isomor-
phisms between complex tori and the corresponding elliptic curves.

(4.7) Theorem. For two equivalent lattices L and L = λL we have the following
commutative diagram where h is defined in (4.3):
λ −1
C/λL −−−−→ C/L
 
 h

h
f

E (C) −−−−→ E(C).
2
3
2
3
The curve E is defined by y = 4x −g 2 x−g 3 ,the curve E by y = 4x −g x−g ,

2 3
and the isomorphism f by the admissible change of variable with u = λ.
2
2
Proof. The relation ℘(λ −1 z, L) = λ ℘(z,λL) corresponds to xf = λ x under
3
3
f , the relation ℘ (λ −1 z, L) = λ ℘ (z,λL) corresponds to yf = λ y under f ,the

4
4
relation g 2 (L) = λ g 2 (λL) corresponds to g 2 = λ g , and the relation g 3 (L) =
2
6
λ g 3 (λL) corresponds to g 3 = λ g . This proves the theorem.
6
3
3 3
(4.8) Remark. From 3(3.5) the discriminant of the cubic x + ax + b is Disc(x +
3
2
3
ax + b) = 27b + 4a and it is also the discriminant of (2x) + a(2x) + b or
3
2
3
3
4x + ax + b/2. Hence Disc(4x − g 2 x − g 3 ) = 4(27g − g ). For a lattice L
3 2
3
2
the corresponding elliptic curve defined by y = 4x − g 2 (L)x − g 3 (L) we defne
2
3
(L) = g 2 (L) − 27g 3 (L) and
g
3 2 (L) 3
j(L) = 12 .
(L)
Then j(L) = j(E) where E is the elliptic curve defined by the Weierstrass equation
2
4
3
12
3
3
3
y = 4x − g 2 (L)x − g 3 (L).Since g 2 (L) = (λ g 2 (λL)) = λ g 2 (λL) and
12
(L) = λ (λL),wehave j(L) = j(λL).
(4.9) Summary Remark. The j-function has the very basic property that it clas-

sifies elliptic curves up to isomorphism, that is, two curves E and E over C are

isomorphic if and only if j(E) = j(E ). This function will also come up in an
essential way in Chapter 11 on modular functions.