Elliptic Curves over C - Elliptic Curves: Number Theory and Cryptography - page 287

℘ ( z ) takes on purely imaginary values. Reasoning similar to that above
(including the same change of variables) yields

1 /k

2 i dt

√ e 3 −

e 1 + √ e 3 −

( t 2

ω 1 =

k 2 t 2 ) .

e 2

− 1)(1 −

1

Let k = √ 1 − k 2 and make the substitution

t =(1 − k 2 u 2 ) − 1 / 2 .

The integral becomes

1

= K ( k )= K ( 1 − k 2 ) .

dt
(1 − t 2 )(1 − k 2 t 2 )

0

Therefore,

e 2 K ( 1 − k 2 ) .

2 i

ω 1 = √ e 3 −

e 1 + √ e 3 −

Therefore, both ω 1 and ω 2 can be expressed in terms of elliptic integrals.

9.4.1 The Arithmetic-Geometric Mean

In this subsection, we introduce the arithmetic-geometric mean. It yields a
very fast and ingenious method, due to Gauss, for computing elliptic integrals.

Start with two positive real numbers a, b . Define a n and b n by

a 0 = a, b 0 = b

a n = 1

2 ( a n− 1 + b n− 1 )

(9.20)

b n = a n− 1 b n− 1 .

Then a n is the arithmetic mean (=average) of a n− 1 and b n− 1 ,and b n is their
geometric mean.