26 1. ALGEBRAIC THEORY OF COMPLEX MULTIPLICATION

Mate(D) is the product of the Z-degrees of L and the centralizer of L in Mate(D).

But L is its own centralizer, so

e2[D

: Z] = dimZ Mate(D) = [L :

Z]2

=

e2(2d/[Z

:

Q])2.

We conclude that 2d/[Z : Q] = [D : Z], so (by Proposition 1.2.3.1) 2d/[Z : Q] is

the common Z-degree of all maximal commutative subfields of the central division

algebra D =

End0(C)

over Z, or equivalently 2d is the Q-degree of all such fields.

But 2d = 2 dim(C), so choosing any maximal commutative subfield of D shows

that C admits suﬃciently many complex multiplications.

1.3.3. CM algebras and CM abelian varieties. The following three conditions

on a number field L are equivalent:

(1) L has no real embeddings but is quadratic over a totally real subfield,

(2) for every embedding j : L → C, the subfield j(L) ⊂ C is stable under complex

conjugation and the involution x →

j−1(j(x))

in Aut(L) is non-trivial and

independent of j,

(3) there is a non-trivial involution τ ∈ Aut(L) such that for every embedding

j : L → C we have j(τ(x)) = j(x) for all x ∈ L.

The proof of the equivalence is easy. When these conditions hold, τ in (3) is unique

and its fixed field is the maximal totally real subfield

L+

⊂ L (over which L is

quadratic). The case

L+

= Q corresponds to the case when L is an imaginary

quadratic field.

1.3.3.1. Definition. A CM field is a number field L satisfying the equivalent

conditions (1), (2), and (3) above. A CM algebra is a product L1 × · · · × Ls of

finitely many CM fields (with s 1).

The reason for this terminology is due to the following important result (along

with Example 1.5.3).

1.3.4. Theorem (Tate). Let A be an abelian variety of dimension g 0 over a

field K. Suppose A admits suﬃciently many complex multiplications. Then there

exists a CM algebra P ⊂

End0(A)

with [P : Q] = 2 dim(A). In case A is isotypic

we can take P to be a CM field.

The proof of this theorem (which ends with the proof of Lemma 1.3.7.1) will

require some effort, especially since we consider an arbitrary base field K. Before

we start the proof, it is instructive to consider an example.

1.3.4.1. Example. Consider A =

E2

with an elliptic curve E over K = C such

that L :=

End0(E)

is an imaginary quadratic field. The endomorphism algebra

End0(A)

= Mat2(L) is simple and contains as its maximal commutative subfields

all quadratic extensions of L. Those extensions which are biquadratic over Q are

CM fields, and the rest are not CM fields. Hence, in the setup of Theorem 1.3.4, even

when A is isotypic and char(K) = 0 there can be maximal commutative semisimple

subalgebras of

End0(A)

that are not CM algebras. However, if char(K) = 0 and A

is simple (over K) then

End0(A)

is a CM field; see Proposition 1.3.6.4.