24 1. ALGEBRAIC THEORY OF COMPLEX MULTIPLICATION
and moreover P is a field of degree 2g over Q then A is isotypic and P is a maximal
commutative subfield of
End0(A).
Proof. Consider the decomposition P = Li into a product of fields. Using
the primitive idempotents of P , we get a corresponding decomposition Ai of A
in the isogeny category of abelian varieties over K, with each Ai
= 0 and each Li
a commutative subfield of
End0(Ai)
compatibly with the inclusion
End0(Ai)

End0(A)
and the equality Li = P . Since dim(A) =

dim(Ai), to prove that
[P : Q] 2g it suffices to treat the Ai’s separately, which is to say that we may
and do assume that P = L is a field.
Since D =
End0(A)
is of finite rank over Q, clearly [L : Q] is finite. Choose
a prime different from char(K). Recall that V (A) denotes Q ⊗Z T (A) for
T (A) := lim

A[
n](Ks).
The injectivity of the natural map
L := Q ⊗Q L EndQ (V (A))
(see Proposition 1.2.5.1) implies that L acts faithfully on the Q -vector space V (A)
of rank 2g. But L =
w|
Lw, where w runs over all -adic places of L, so each
corresponding factor module V (A)w over Lw is non-zero as a vector space over Lw.
Hence,
2g = dimQ V (A) =
w|
dimQ V (A)w
w|
[Lw : Q ] = [L : Q]
with equality if and only if V (A) is free of rank 1 over L .
Assume that equality holds, so V (A) is free of rank 1 over L . If A is not isotypic
then by passing to an isogenous abelian variety we may arrange that A = B × B
with B and B non-zero abelian varieties such that Hom(B, B ) = 0 = Hom(B , B).
Hence,
End0(A)
=
End0(B)
×
End0(B
) and so L embeds into
End0(B).
But
2 dim(B) 2 dim(A) = [L : Q], so we have a contradiction (since B = 0).
It remains to prove, without assuming P is a field, that if [P : Q] = 2g then
P is its own centralizer in
End0(A).
(In case P is a field, so A is isotypic and
hence
End0(A)
is simple, such a centralizer property would imply that P is a
maximal commutative subfield of
End0(A),
as desired.) Consider once again the
ring decomposition P = Li and the corresponding isogeny decomposition Ai
of A as at the beginning of this proof. We have [Li : Q] 2 dim(Ai) for all i,
and these inequalities add up to an equality when summed over all i, so in fact
[Li : Q] = 2 dim(Ai) for all i. The preceding analysis shows that each V (Ai) is
free of rank 1 over Li, := Q ⊗Q Li, and so likewise V (A) is free of rank 1 over
P . Hence, EndP (V (A)) = P , so if Z(P ) denotes the centralizer of P in
End0(A)
then the P -algebra map
Z(P ) = Q ⊗Q Z(P ) EndQ (V (A))
is injective (Proposition 1.2.5.1) and lands inside EndP (V (A)) = P . In other
words, the inclusion P Z(P ) of Q-algebras becomes an equality after scalar
extension to Q , so P = Z(P ) as desired.
The preceding theorem justifies the interest in the following concept.
1.3.1.2. Definition. An abelian variety A of dimension g 0 over a field K
admits sufficiently many complex multiplications (over K) if there exists a commu-
tative semisimple Q-subalgebra P in
End0(A)
with rank 2g over Q.
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