Complex Multiplication - Elliptic Curves: Number Theory and Cryptography - page 313

Therefore, multiplication by β on C /L corresponds to the map

( x, y ) → ( R ( x ) ,yS ( x ))

on E . This is precisely the statement that β induces an endomorphism of E .

Theorem 10.1 imposes rather severe restrictions on the endomorphism ring
of E . We'll show below that End( E ) is either Z or an order in an imaginary
quadratic field. First, we need to say what this means. We'll omit the proofs
of the following facts, which can be found in many topics on algebraic number
theory. Let d> 0 be a squarefree integer and let
K = Q ( √ −d )= {a + b √ −d | a, b ∈ Q }.

Then K is called an imaginary quadratic field . The largest subring of K
that is also a finitely generated abelian group is

⎨

Z 1+ √ − d
2

if d ≡ 3(mod )

O K =

⎩

Z √ −d if d ≡ 1 , 2(mod ,

where, in these two cases, Z [ δ ]= {a + bδ | a, b ∈ Z } .An order in an imaginary
quadratic field is a ring R such that Z ⊂ R ⊆O K and Z = R . Suchanorder
is a finitely generated abelian group and has the form

R = Z + Z fδ,
where f> 0andwhere δ =(1+ √ − d ) / 2or √ − d , corresponding respectively

to the two cases given above. The integer f is called the conductor of R and
is the index of R in O K .The discriminant of R is
D R = −

f 2 d if d ≡ 3(mod )

− 4 f 2 d

if d ≡ 1 , 2(mod .

It is the discriminant of the quadratic polynomial satisfied by fδ .
A complex number β is an algebraic integer if it is a root of a monic
polynomial with integer coecients. The only algebraic integers in Q are the
elements of Z .If β is an algebraic integer in a quadratic field, then there are
integers b, c such that β 2 + bβ + c = 0. The set of algebraic integers in an
imaginary quadratic field K is precisely the ring O K defined above. An order
is therefore a subring (not equal to Z ) of the ring of algebraic integers in K .
If β ∈ C is an algebraic number (that is, a root of a polynomial with rational
coe cients), then there is an integer u =0suchthat uβ is an algebraic integer.

THEOREM 10.2
Let E be an elliptic curve over C .ThenEnd ( E ) isisom orphiceither to Z
or toanorderinan maginary quadraticfie d.