Elliptic Curves over C - Elliptic Curves: Number Theory and Cryptography - page 293

elliptic curves with com p lex multiplication in terms of values of the gamma
function (see [101]).

There are also formulas similar to those of Theorem 9.26 for the case where
g 2 ,g 3 ∈ R but 4 x 3 − g 2 x − g 3 has only o ne real root. Let e 1 be the unique

− g 2 x − g 3 and let e = 3 e 1 − (1 / 4) g 2 .Then

real root of 4 x 3

2 π
M ( √ 4 e , √ 2 e +3 e 1 )

ω 1 = (9.23)

ω 1
2

πi
M ( √ 4 e , √ 2 e − 3 e 1 ) .

ω 2 = − + (9.24)

The proof is similar to the one when there are three real roots.
For more on the arithmetic-geometric mean, including how it has been used
to compute π very accurately and how it behaves for complex arguments, see
[17] and [30].

9.5 Division Polynomials

In this section, we prove Theorem 3.6, which gives a formula for n ( x, y ),
where n> 1 is an integer and ( x, y ) is a point on an elliptic curve. We'll
start with the case of an elliptic curve in characteristic zero, then use this to
deduce the case of positive characteristic.
Let E be an elliptic curve over a field of characteristic 0, given by an
equation y 2 = x 3 + Ax + B . All of the equations describing the group law
are defined over Q ( A, B ). Since C is algebraically closed and has infinite
transcendence degree over Q , it is easy to see that Q ( A, B ) may be considered
as a subfield of C . Therefore, we regard E as an elliptic curve defined over
C . By Theorem 9.21, there is a lattice L corresponding to E .Let ℘ ( z )bethe
associated Weierstrass ℘ -function, which satisfies the relation

( ℘ ) 2 =4 ℘ 3 − g 2 ℘ − g 3 ,

4 B . We'll derive formulas for ℘ ( nz )and ℘ ( nz ), then
use x = ℘ ( z )and y = ℘ ( z ) / 2 to obtain the desired formulas for n ( x, y ).

with g 2 = − 4 A, g 3 = −

LEMMA 9.27
Thereis a doubly periodicfunction f n ( z ) su ch that

f n ( z ) 2 = n 2 ( ℘ ( z ) − ℘ ( u )) .

0 = u ∈ ( C /L )[ n ]

Thesign of f n can be chosen so that