Complex Multiplication - Elliptic Curves: Number Theory and Cryptography - page 314

PROOF Let L = Z ω 1 + Z ω 2 be the lattice corresponding to E ,andlet

R = {β ∈ C | βL ⊆ L}.

It is easy to see that Z ⊆ R and that R is closed under addition, subtraction,
and multiplication. Therefore, R is a ring. Suppose β ∈ R . There exist

integers j, k, m, n such that

βω 1 = jω 1 + kω 2 , βω 2 = mω 1 + nω 2 .

Then

β − j −k
−mβ− n

ω 1
ω 2

=0 ,

so the determinant of the matrix is 0. This implies that

β 2 − ( j + n ) β +( jn − km )=0 .

Since j, k, m, n are integers, this means that β is an algebraic integer, and
that β lies in some quadratic field K .
Suppose β kω 2 = 0 gives a dependence relation

between ω 1 and ω 2 with real coecients. Since ω 1 and ω 2 are linearly inde-
pendent over R ,wehave β = j ∈ Z . Therefore, R ∩ R = Z .
Suppose now that R = Z .Let β ∈ R with β ∈ Z .Then β is an algebraic
integer in a quadratic fiel d K .Since β ∈ R , the field K must be imaginary

∈ R .Then( β − j ) ω 1 −

quadratic, say K = Q ( √ −d ). Let β ∈ Z be another element of R .Then

β ∈ K = Q ( √ −d )forsome d .Since β + β also must lie in a quadratic
field, it follows (see Exercise 10.1) that K = K . Therefore, R ⊂ K , and since

all elements of R are algebraic integers, we have

R ⊆O K .

Therefore, if R = Z ,then R is an order in an imaginary quadratic field.

Example 10.1
Let E be y 2 =4 x 3 − 4 x . We showed at the beginning of this section that

Z [ i ] End( E ). Since End( E )isanorderin Q ( i ) and every such order is

contained in the ring Z [ i ] of algebraic integers in Q ( i ), we must have

⊆

End( E )= Z [ i ] .

Suppose from now on that E has complex multiplication, which means that
R = End( E ) is an order in an imaginary quadratic field K . Rescaling L does
not change R , so we may consider

ω − 1 L = Z + Z τ,

2