Complex Multiplication - Elliptic Curves: Number Theory and Cryptography - page 315

ω − 1 with τ ∈H = { z ∈ C | ( z ) > 0 } .Let β ∈ R with β ∈ Z .Since1 ∈ L ,we

2

have β · 1= m · 1+ nτ with m, n ∈ Z and n = 0. Therefore,

τ =( β − m ) /n ∈ K. (10.1)

Let u be an integer such that uτ R . Such an integer exists since τ multiplied

by n is in O K ,and R is of finite index in O K .Then

∈

L = uω − 2 L = Z u + Z uτ ⊆ R.

Then L is a nonempty subset of R that is closed under addition and sub-
traction, and is closed under multiplication by elements of R (since L is a
rescaling of L ). This is exactly what it means for L to be an ideal of R .We
have proved the first half of the following.

PROPOSITION 10.3
Let R be an order inan maginary quadraticfie d. Let L be a lattice su ch
that R = End ( C /L ) .Thenthere exists γ

∈ C × su ch that γL isanideal of
R . C onversely, if L is a subset of C and γ ∈ C × issuchthat γL isanideal
of R ,then L isalattice an d R ⊆ End ( C /L ) .

PROOF By End( C /L ), we mean End( E ), where E is the elliptic curve
corresponding to L under Theorem 9.10.
We proved the first half of the proposition above. For the converse, assume
that γL is an ideal of R .Let0 = x ∈ γL .Then

Rx ⊆ γL ⊆ R.

Since R and therefore also Rx are abelian groups of rank 2 (that is, isomorphic
to Z ⊕ Z ), the same must be true for γL . This means that there exist ω 1 ,ω 2 ∈ L

such that

γL = γ Z ω 1 + γ Z ω 2 .
Since R contains two elements linearly independent over R ,sodoes Rx ,and
therefore so does L . It follows that ω 1 and ω 2 are linearly independent over
R . Therefore, L = Z ω 1 + Z ω 2 is a lattice. Since γL is an ideal of R ,wehave
RγL ⊆ γL , and therefore RL ⊆ L . Therefore R ⊆ End( C /L ).

Note that sometimes R is not all of End( C /L ). For example, suppose

R = Z [2 i ]= { a +2 bi | a, b ∈ Z } and let L = Z [ i ]. Then R is an order in Q ( i )

and RL = R .

We say that two lattices L 1 ,L 2 are homothetic if there exists γ ∈ C ×
such that γL 1 = L 2 . We say that two ideals I 1 ,I 2 of R are equivalent if there
exists λ ∈ K × such that λI 1 = I 2 . Regard I 1 and I 2 as lattices, and suppose
I 1 and I 2 are homothetic. Then γI 1 = I 2 for some γ .Chooseany x =0in I 1 .

⊆ L , but End( C /L )= Z [ i ]