Complex Multiplication - Elliptic Curves: Number Theory and Cryptography - page 319

Example 10.2
Let E be the elliptic curve y 2 = x 3 − x . It has good reduction for all primes

p = 2. The endomorphism ring R of E is Z [ i ], where

i ( x, y )=( −x, iy )
(see Section 10.1). This endomorphism ring is contained in Q ( √ − 4), where
we use −D = − 4 since it is the discriminant of R . We know that − 4isa
square mod an odd prime p if and only if p ≡ 1 (mod 4). Therefore, E mod
p is ordinary if and only if p ≡ 1 (mod 4). This is exactly what we obtained
in Proposition 4.37.
When p 3 (mod 4), it is easy to see that the endomorphism ring of E

mod p is noncommutative. Since i p =

≡

− i ,wehave

φ p ( i ( x, y )) = φ p ( −x, iy )=( −x p , −iy p ) ,

and

i ( φ p ( x, y )) = i ( x p ,y p )=( −x p ,iy p ) .

Therefore,

iφ p = −φ p i,

so i and φ p do not commute.

The following result, known as Deuring's Lifting Theorem ,showsthat
the method given in Theorem 10.7 for obtaining ordinary elliptic curves mod
p with complex multiplication is essentially the only way. Namely, it implies
that an elliptic curve with complex multiplication over a finite field can be
obtained by reducing an elliptic curve with complex multiplication in charac-
teristic zero.

THEOREM 10.8
Let E be an elliptic curve defined over a finitefie dand et α be an endo-
m orphism of E .Thenthere existsanellipticcurve E defined over a finite
extension K of Q and an endom orphism α of E su ch that E isthe reduction
of E modsomeprimeideal of the ring of algebraic ntegers of K and the
red u ction of α is α .

For a proof in the ordinary case, see [70, p. 184].
It is not possible to extend the theorem to lifting two arbitrary endomor-
phisms simultaneously. For example, the endomorphisms i and φ p in the
above example cannot be simultaneously lifted to characteristic 0 since they
do not commute. All endomorphism rings in characteristic 0 are commutative.
Finally, we give an example of a supersingular curve in characteristic 2. In
particular, we'll show how to identify the maximal order of H in the endo-
morphism ring.