1.3. COMPLEX MULTIPLICATION 25

The reason for the terminology in Definition 1.3.1.2 is due to certain examples

with K = C and P a number field such that the analytic uniformization of A(C)

expresses the P -action in terms of multiplication of complex numbers; see Example

1.5.3. The classical theory of complex multiplication focused on the case of Defini-

tion 1.3.1.2 in which P is a field, but it is useful to allow P to be a product of several

fields (i.e., a commutative semisimple Q-algebra). For example, by Theorem 1.3.1.1

this is necessary if we wish to consider the theory of complex multiplication with A

that is not isotypic, or more generally if we want Definition 1.3.1.2 to be preserved

under the formation of products. The theory of Shimura varieties provides further

reasons not to require P to be a field.

Note that we do not consider A to admit suﬃciently many complex multipli-

cations merely if it does so after an extension of the base field K.

1.3.2. Example. The elliptic curve y2 = x3 − x admits suﬃciently many complex

multiplications over Q(

√

−1) but not over Q. More generally,

End0(E)

= Q for

every elliptic curve E over Q (since the tangent line at the origin is too small to

support a Q-linear action by an imaginary quadratic field), so in our terminology

an elliptic curve over Q does not admit suﬃciently many complex multiplications.

1.3.2.1. Proposition. Let A be a non-zero abelian variety over a field K. The

following are equivalent.

(1) The abelian variety A admits suﬃciently many complex multiplications.

(2) Each isotypic part of A admits suﬃciently many complex multiplications.

(3) Each simple factor of A admits suﬃciently many complex multiplications.

See Definition 1.2.5.2 for the terminology used in (2).

Proof. Let {Bi} be the set of isotypic parts of A, so

End0(Bi)

Matei

(End0(Ci))

where Ci is the unique simple factor of Bi and ei 1 is its multiplicity as such.

Since

End0(A)

=

End0(Bi),

(2) implies (1). It is clear that (3) implies (2).

Conversely, assume that

End0(A)

contains a Q-algebra P satisfying [P : Q] =

2 dim(A). There is a unique decomposition P = Li with fields L1,...,Ls, and

∑

[Li : Q] = 2 dim(A). We saw in the proof of Theorem 1.3.1.1 that by replacing

A with an isogenous abelian variety we may arrange that A = Ai with each Ai

a non-zero abelian variety having Li ⊂

End0(Ai)

compatibly with the embedding

End0(Ai)

⊂

End0(A)

and the equality Li = P . Thus, [Li : Q] 2 dim(Ai) for

all i (by Theorem 1.3.1.1), and adding this up over all i yields an equality, so each

Ai admits suﬃciently many complex multiplications using Li. Since each simple

factor of A is a simple factor of some Ai, to prove (3) we are therefore reduced to

the case when P = L is a field.

Applying Theorem 1.3.1.1 once again, L is its own centralizer in

End0(A)

and

A is isotypic, say with unique simple factor C appearing with multiplicity e. In

particular,

End0(A)

= Mate(D) for the division algebra D =

End0(C)

of finite

rank over Q. If Z denotes the center of D then D is a central division algebra

over Z, and L contains Z since L is its own centralizer in

End0(A)

= Mate(D).

Letting d = dim(C), Mate(D) contains the maximal commutative subfield L of

degree 2g/[Z : Q] = (2d/[Z : Q])e over Z.

As we noted in the proof of Proposition 1.2.3.1 (parts of which are carried

out for central simple algebras that may not be division algebras), the Z-degree of