Complex Multiplication - Elliptic Curves: Number Theory and Cryptography - page 317

where a, b, c are integers and a = 0. It follows that multiplication by aτ maps

the lattice L τ = Z τ + Z into itself (for example, aτ · τ = − bτ − c ∈ L τ ).

Therefore, C /L τ has complex multiplication. This proves (1).
Suppose τ = Z ,

so R is an order in K . By Proposition 10.4, j ( τ ) is algebraic. This proves (2).

∈ K .Let R be the endomorphism ring of C /L τ .By(1), R

10.2 Elliptic Curves over Finite Fields

An elliptic curve E over a finite field F q always has complex multiplication.
In most cases, this is easy to see. The Frobenius endomorphism φ q is a root
of

X 2 − aX + q =0 ,

2 √ q .If < 2 √ q , then this polynomial has only complex roots, where | a |≤ | a |

so φ q ∈ Z . Therefore,

Z = Z [ φ q ] ⊆ End( E ) .

2 √ q , the ring of endomorphisms is still larger than Z ,sothere
is complex multiplication in this case, too. In fact, as we'll discuss below, the
endomorphism ring is an order in a quaternion algebra, hence is larger than
an order in a quadratic field.
Recall the Hamiltonian quaternions

When a = ±

H = { a + b i + c j + d k | a, b, c, d ∈ Q } ,

where i 2 = j 2 = k 2

ji . This is a noncommutative
ring in which every nonzero element has a multiplicative inverse. If we allow
the coe cients a, b, c, d to be real numbers or 2-adic numbers, then we still
obtain a ring where every nonzero element has an inverse. However, if a, b, c, d
are allowed to be p -adic numbers (see Appendix A), where p is an odd prime,
then the ring contains nonzero elements whose product is 0 (see Exercise 10.4).
Such elements cannot have inverses. Corresponding to whether there are zero
divisors or not, we say that H is split at all odd primes and is ramified at
2and ∞ (this use of ∞ is the common way to speak about the real numbers
when simultaneously discussing p -adic numbers; see Section 8.8).
In general, a definite quaternion algebra is a ring of the form

= − 1and ij = k = −

Q = {a + bα + cβ + dαβ | a, b, c, d ∈ Q },

where

∈ Q , α 2 < 0 , β 2 < 0 , βα = −αβ
(“definite” refers to the requirement that α 2 < 0and β 2 < 0). In such a ring,
every nonzero element has a multiplicative inverse (see Exercise 10.5). If this

α 2 ,β 2