Complex Multiplication - Elliptic Curves: Number Theory and Cryptography - page 312

for all z 1 ,z 2 ∈ U . By subtracting an appropriate element of L ,wemay

assume that α (0) = 0. By continuity, α ( z ) is near 0 when z is near 0. If U is
su ciently small, we may therefore assume that

α ( z 1 + z 2 )= α ( z 1 )+ α ( z 2 )

for all z 1 ,z 2 ∈ U (since both sides are near 0, they can differ only by the
element 0 ∈ L ). Therefore, for z ∈ U ,wehave

α ( z + h ) − α ( z )

α ( z ) = lim
h

h

→ 0

α ( z )+ α ( h ) − α ( z )
h

= lim
h → 0

α ( h ) − α (0)

= α (0) . = lim

h

h

→ 0

Let β = α (0). Since α ( z )= β for all z ∈ U ,wemusthave

α ( z )= βz

for all z ∈ U .
Now let z ∈ C be arbitrary. There exists an integer n such that z/n ∈ U .
Therefore,

α ( z ) ≡ nα ( z/n )= n ( βz/n )= βz (mod L ) ,

so the endomorphism α is given by multiplication by β .Since α ( L ) ⊆ L ,it
follows that

βL ⊆ L.

We have proved half of the following.

THEOREM 10.1
Let E be an elliptic curve over C corresponding tothe lattice L .Then

End ( E ) {β ∈ C | βL ⊆ L}.

PROOF We have shown that all endomorphisms are given by numbers
β . We need to show that all such β 's give endomorphisms. Suppose β ∈ C
satisfies βL ⊆ L . Then multiplication by β gives a homomorphism

β : C /L → C /L.

We need to show that the corresponding map on E is given by rational func-
tions in x, y .
The functions ℘ ( βz )and ℘ ( βz ) are doubly periodic with respect to L ,since
βL ⊆ L . By Theorem 9.3, there are rational functions R and S such that

℘ ( βz )= ℘ ( z ) S ( ℘ ( z )) . ℘ ( βz )= R ( ℘ ( z )) ,