Complex Multiplication - Elliptic Curves: Number Theory and Cryptography - page 311

Therefore, the map

z → iz

gives a map

( x, y )=( ℘ ( z ) ,℘ ( z )) ( ℘ ( iz ) ,℘ ( iz )) = ( → − x, iy ) .

This is a homomorphism from E ( C )to E ( C ) and it is clearly given by rational
functions. Therefore, it is an endomorphism of E , as in Section 2.9. Let

Z [ i ]= {a + bi | a, b ∈ Z }.

Then Z [ i ] is a ring, and multiplication by elements of Z [ i ] sends L into it-
self. Correspondingly, if a + bi ∈ Z [ i ]and( x, y ) ∈ E ( C ), then we obtain an
endomorphism of E defined by

( x, y ) → ( a + bi )( x, y )= a ( x, y )+ b ( −x, iy ) .

Since multiplication by a and b can be expressed by rational functions, mul-
tiplication of points by a + bi is an endomorphism of E , as in Section 2.9.
Therefore,

Z [ i ] ⊆ End( E ) ,

where End( E ) denotes the ring of endomorphisms of E . (We'll show later
that this is an equality.) Therefore, End( E ) is strictly larger than Z ,so E
has complex multiplication. Just as Z [ i ] is the motivating example for a lot
of ring theory, so is E the prototypical example for complex multiplication.
We now consider endomorphism rings of arbitrary elliptic curves over C .
Let E be an elliptic curve over C , corresponding to the lattice

L = Z ω 1 + Z ω 2 .

Let α be an endomorphism of E . Recall that this means that α is a homo-
morphism from E ( C )to E ( C ), and that α is given by rational functions:

α ( x, y )=( R ( x ) ,yS ( x ))

for rational functions R, S .Themap

Φ( z )=( ℘ ( z ) ,℘ ( z ))

Φ: C /L → E ( C ) ,

(see Theorem 9.10) is an isomorphism of groups. The map

α ( z )=Φ − 1 ( α (Φ( z )))

is therefore a homomorphism from C /L to C /L . If we restrict to a suciently
small neighborhood U of z =0,weobtainananalyticmapfrom U to C such
that

α ( z 1 + z 2 ) ≡ α ( z 1 )+ α ( z 2 )(mod L )