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.