1.3. COMPLEX MULTIPLICATION 27

1.3.5. We will begin the proof of Theorem 1.3.4 now, but at a certain point we will

need to use deeper input concerning the fine structure of endomorphism algebras of

simple abelian varieties over general fields. At that point we will digress to review

the required structure theory, and then we will complete the argument.

By Proposition 1.3.2.1, every simple factor of A admits suﬃciently many com-

plex multiplications. Thus, to prove the existence of the CM subalgebra P in Theo-

rem 1.3.4 it suﬃces to treat the case when A is simple. Note that in the simple case

such a CM subalgebra is automatically a field, since the endomorphism algebra is

a division algebra. Let us first show that the result in the simple case implies that

in the general isotypic case we can find P as a CM field. For isotypic A, by passing

to an isogenous abelian variety we can arrange that A = A

m

for a simple abelian

variety A over K and some m 1. Thus, if g = dimA then g = mg and

End0(A

)

contains a CM field P of degree 2g over Q. But

End0(A) Matm(End0(A

)) and

this contains Matm(P ). To find a CM field P ⊂

End0(A)

of degree 2g = 2g m over

Q it therefore suﬃces to construct a degree-m extension field P of P such that P

is a CM field.

Let P

+

be the maximal totally real subfield of P , so for any totally real field

P + of finite degree over P

+

the ring P = P + ⊗P

+

P is a field quadratic over P +

and it is totally complex, so it is a CM field and clearly [P : Q] = [P : P ][P : Q] =

2g [P

+

: P

+

]. Hence, to find the required CM field P in the isotypic case it suﬃces

to construct a degree-m totally real extension of P

+

. To do this, first recall the

following basic fact from number theory [15, §6]:

1.3.5.1. Theorem (weak approximation). For any number field L and finite set

S of places of L, the map L →

v∈S

Lv has dense image.

Proof. This is [15, §6].

Applying this to P

+

, we can construct a monic polynomial f of degree m in

P

+

[u] that is very close to a totally split monic polynomial of degree m at each

real place and is very close to an irreducible (e.g., Eisenstein) polynomial at a single

non-archimedean place. It follows that f is totally split at each real place of P

+

and is irreducible over P

+

, so the ring P + = P

+

[u]/(f) is a totally real field of

degree m over P

+

as required.

1.3.5.2. We may and do assume for the remainder of the argument that A is

simple. In this case D =

End0(A)

is a central division algebra over a number field

Z, so the commutative semisimple Q-subalgebra P ⊂ D is a field, and the proof of

Proposition 1.3.2.1 shows that the common Q-degree of all maximal commutative

subfields of D is 2g. Hence, our problem is to construct a maximal commutative

subfield of D that is a CM field.

Let TrdD/Q = TrZ/Q ◦ TrdD/Z, where TrdD/Z is the reduced trace. An abelian

variety over any field admits a polarization, so choose a polarization of A over K.

Let x →

x∗

denote the associated Rosati involution on D (so

(xy)∗

=

y∗x∗

and

x∗∗ = x).

1.3.5.3. Lemma. The quadratic form x → TrdD/Q(xx∗) on D is positive-definite.

Proof. For any central simple algebra D over any field K whatsoever, let n =

[D : K] and define the variant TrmD/K : D → K of the reduced trace to be the