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 settles the case of characteristic 0,
and Proposition gives that D =
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. Lemma (Tate). Let D be a central division algebra of degree
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, 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
Dv := Zv ⊗Z D
of rank
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, 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
such that for each
place v0 of
beneath a place v Σ, the extension Z
has a unique place
v0 over v0 and is totally ramified (resp. unramified) at v0 when
is unramified
(resp. ramified) at v. Hence, (Z
)v0 and Zv are linearly disjoint over
. 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 This also finishes the proof of Theorem 1.3.4. Corollary. An isotypic abelian variety A with sufficiently many complex
multiplications remains isotypic after any extension of the base field.
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