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.
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