1.3. COMPLEX MULTIPLICATION 31

1.3.7. Returning to the proof of Theorem 1.3.4, recall that we reduced the proof

to the case of simple A. Proposition 1.3.6.4(1) settles the case of characteristic 0,

and Proposition 1.3.6.4(2) gives that D =

End0(A)

is an Albert algebra of Type

III or IV when char(K) 0. If D is of Type III then the center Z is totally real

and d is even, whereas if D is of Type IV then Z is CM. Thus, we can apply the

following general lemma to conclude the proof.

1.3.7.1. Lemma (Tate). Let D be a central division algebra of degree

d2

over a

number field Z that is totally real or CM. If Z is totally real then assume that d is

even. There exists a maximal commutative subfield L ⊂ D that is a CM field.

The parity condition on d is necessary when Z is totally real, since d = [L : Z]

by maximality of L in D.

Proof. By Proposition 1.2.3.1, any degree-d extension of Z that splits D is

a maximal commutative subfield of D. Hence, we just need to find a degree-d

extension L of Z that is a CM field and splits D. Let Σ be a finite non-empty

set of finite places of Z containing the finite places at which D is non-split. By

the structure of Brauer groups of local fields, for any v ∈ Σ the central simple

Zv-algebra

Dv := Zv ⊗Z D

of rank

d2

over Zv is split by any extension of Zv of degree d.

First assume that Z is totally real, so d is even. By weak approximation

(Theorem 1.3.5.1), there is a monic polynomial f over Z of degree d/2 that is

close to a monic irreducible polynomial of degree d/2 over Zv for all v ∈ Σ (and in

particular f is irreducible over all such Zv, and hence over Z since Σ is non-empty).

We can also arrange that for each real place v of Z the polynomial f viewed over

Zv R is close to a totally split monic polynomial of degree d/2 and hence is

totally split over Zv. Thus, Z := Z[u]/(f) is a totally real extension field of Z

with degree d/2. By the same method, we can construct a quadratic extension

L/Z that is unramified quadratic over each place v over a place in Σ and is also

totally complex (by using approximations to irreducible quadratics over R at the

real places of Z ). This L is a CM field and it is designed so that Zv ⊗Z L is a

degree-d field extension of Zv for all v ∈ Σ. Hence, DL is split at all places of L

(the archimedean ones being obvious), so DL is split.

Assume next that Z is a CM field. Let Z+ ⊂ Z be the maximal totally real

subfield. By the same weak approximation procedure as above (replacing d/2 with

d), we can construct a degree d totally real extension Z

+

/Z+

such that for each

place v0 of

Z+

beneath a place v ∈ Σ, the extension Z

+

/Z+

has a unique place

v0 over v0 and is totally ramified (resp. unramified) at v0 when

Z/Z+

is unramified

(resp. ramified) at v. Hence, (Z

+

)v0 and Zv are linearly disjoint over

(Z+)v0

. We

conclude that Z

+

and Z are linearly disjoint over Z , so L := Z

+

⊗Z+ Z is a field

and each v ∈ Σ has a unique place w over it in L. Clearly [Lw : Zv] = d for all

such w, so L splits D. By construction, L is visibly CM. We have proved Lemma

1.3.7.1. This also finishes the proof of Theorem 1.3.4.

1.3.7.2. Corollary. An isotypic abelian variety A with suﬃciently many complex

multiplications remains isotypic after any extension of the base field.