Complex Multiplication - Elliptic Curves: Number Theory and Cryptography - page 310

Chapter 10

Complex Multiplication

The endomorphisms of an elliptic curve E always include multiplication by
arbitrary integers. When the endomorphism ring of E is strictly larger than Z ,
we say that E has complex multiplication . As we'll see, elliptic curves over
C with complex multiplication correspond to lattices with extra symmetry.
Over finite fields, all elliptic curves have complex multiplication, and often the
Frobenius provides one of the additional endomorphisms. In general, elliptic
curves with complex multiplication form an interesting and important class
of elliptic curves, partly because of their extra structure and partly because
of their frequent occurrence.

10.1 Elliptic Curves over C
Consider the elliptic curve E given by y 2 =4 x 3

− 4 x over C . Aswesaw
in Section 9.4, E corresponds to the torus C /L ,where L = Z ω + Z iω ,for
a certain ω ∈ R .Since L is a square lattice, it has extra symmetries. For
example, rotation by 90 ◦ sends L into itself. This can be expressed by saying
that iL = L . Using the definition of the Weierstrass ℘ -function, we easily see
that

ω 2

( iz ) 2 +
ω

1 1

( iz − ω ) 2 −

1

℘ ( iz )=

=0

( iω ) 2 ( iz ) 2 +

iω

1 1 1

=

iω ) 2 −

( iz −

=0

= − ℘ ( z ) .

Differentiation yields

℘ ( iz )= i℘ ( z ) .

On the elliptic curve E , we obtain the endomorphism given by

i ( x, y )=( −x, iy ) .