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, 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
for a simple abelian
variety A over K and some m 1. Thus, if g = dimA then g = mg and
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
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]: Theorem (weak approximation). For any number field L and finite set
S of places of L, the map L
Lv has dense image.
Proof. This is [15, §6].
Applying this to P
, we can construct a monic polynomial f of degree m in
[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. We may and do assume for the remainder of the argument that A is
simple. In this case D =
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 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
denote the associated Rosati involution on D (so
x∗∗ = x). 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
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