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
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