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§6. Periods Associated with Elliptic Curves: Elliptic Integrals 185

( π/2
dθ 1 1
= + = π F , , 1; λ by (5.9).
2
0 1 − λ sin θ 2 2
Next we calculate the integral for ω 1 (λ) using the change of variable y =−x + 1or
x = 1 − y

dx
( 1
ω 1 (λ) = √
x(x − 1)(x − λ)
−∞
∞
dy
(
= √
1 −(y − 1)(−y)((y − (1 − λ))
∞ dy
(
= i √
1 y(y − 1)(y − (1 − λ))
1 1

= iπ F , , 1; 1 − λ
2 2
by the first change of variable used to calculate ω 2 (λ). This proves the theorem.
Thus we have shown how to pass from tori to elliptic curves with the elliptic
function ℘ and from elliptic curves over C to tori with the hypergeometric function
F(1/2, 1/2, 1; λ).
There is one case where the calculations of the periods are particularly agreeable,
namely λ = 1/2, −1, or 2. Since for λ = 1/2, 1 − λ = 1/2, we have ω 1 (1/2) =
iω 2 (1/2) and the lattice is of the form Z[i] · & = (Z · i + Z) · &. To determine & for
one of these curves all isomorphic over C, we calculate

( 1 ( π/2
dt dθ
ω 2 (−1) = & = + = √
2
0 t(1 − t ) 0 sin θ
( π/2
= sin 2(1/4)−1 θ cos 2(1/2)−1 θ dθ
0

1
1
4 2
= by (5.4).

3
4
2
3
(6.2) Proposition. The period lattice for the elliptic curve given by y = x − xis
Z[i] · &, where

1
√ 4
& = π .

3
4
As we have remarked before, with elliptic functions we map from a torus to a
2
cubic curve, and with the hypergeometric function we can assign to a cubic y =
x(x − 1)(x − λ) the corresponding torus. The projection w : x : y to w : x from the
cubic curve given by