The reason for the terminology in Definition 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 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
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 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,
= 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. 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 for the terminology used in (2).
Proof. Let {Bi} be the set of isotypic parts of A, so
where Ci is the unique simple factor of Bi and ei 1 is its multiplicity as such.
(2) implies (1). It is clear that (3) implies (2).
Conversely, assume that
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 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
compatibly with the embedding

and the equality Li = P . Thus, [Li : Q] 2 dim(Ai) for
all i (by Theorem, 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 once again, L is its own centralizer in
A is isotypic, say with unique simple factor C appearing with multiplicity e. In
= Mate(D) for the division algebra D =
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
= 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 (parts of which are carried
out for central simple algebras that may not be division algebras), the Z-degree of
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