28 1. ALGEBRAIC THEORY OF COMPLEX MULTIPLICATION
map that sends each y D to the trace of the K-linear map my : D D defined by
d yd. We have TrmD/K = n · TrdD/K, as may be checked by extending scalars
to Ks and directly computing with elementary matrices (see 1.2.3.3). Hence, in
the setting of interest with D =
End0(A)
and K = Z we see that it is equivalent
to prove positive-definiteness for the quadratic form x
TrmD/Q(xx∗),
where
TrmD/Q = TrZ/Q TrmD/Z. The positive-definiteness for TrmD/Q can be verified
by replacing D with EndK
0
(AK), to which [82, §21, Thm. 1] applies.
Lemma 1.3.5.3 says that x x∗ is a positive involution of D (relative to the
linear form TrdD/Q). The existence of such an involution severely constrains the
possibilities for D. First we record the consequences for the center Z.
1.3.5.4. Lemma. The center Z of D =
End0(A)
is either totally real or a CM
field, and in the latter case its canonical complex conjugation is induced by the
Rosati involution defined by any polarization of A over K.
Proof. Fix a polarization and consider the associated Rosati involution x x∗
on the center Z of D. Clearly Z is stable under this involution. The positive-
definite TrdD/Q(xx∗) on D restricts to [D : Z] · TrZ/Q(xx∗) on Z, so TrZ/Q(xx∗)
is positive-definite on Z. If x∗ = x for all x Z then the rational quadratic
form TrZ/Q(x2) is positive-definite on Z, so by extending scalars to R we see that
Tr(R⊗QZ)/R(x2)
is positive-definite. This forces the finite ´ etale R-algebra R ⊗Q Z to
have no complex factors. Hence, Z is a totally real field in such cases.
It remains to show that if the involution x
x∗
is non-trivial on Z for some
choice of polarization then Z is a CM field (so the preceding argument would imply
that the Rosati involution arising from any polarization of A is non-trivial on Z)
and its intrinsic complex conjugation is equal to this involution on Z. Let
Z+
be
the subfield of fixed points in Z for this involution, so [Z :
Z+]
= 2 and 2 TrZ+/Q
is the restriction to Z+ of TrZ/Q. Hence, TrZ+/Q(x2) is positive-definite on Z+,
so
Z+
is totally real. We aim to prove that Z has no real places, so we assume
otherwise and seek a contradiction.
Let v be a real place of Z. Since the involution x
x∗
is non-trivial on Z and
the field Zv R has no non-trivial field automorphisms, the real place v on Z is not
fixed by the involution x
x∗.
Thus, the real place
v∗
obtained from v under the
involution is a real place of Z distinct from v, and so the positive-definiteness
of
TrZ/Q(xx∗)
implies (after scalar extension to R) the positive-definiteness of
Tr(Zv×Zv∗
)/R(xx∗),
where x
x∗
on Zv × Zv∗ = R × R is the involution that
swaps the factors. In other words, this is the quadratic form (c, c ) 2cc , which
by inspection is not positive-definite.
1.3.6. Albert’s classification. To go further with the proof of Theorem 1.3.4,
we need to review properties of endomorphism algebras of simple abelian varieties
over arbitrary fields.
1.3.6.1. Definition. An Albert algebra is a pair consisting of a division algebra D
of finite dimension over Q and a positive involution x
x∗
on D.
For any Albert algebra D and any algebraically closed field K, there exists a
simple abelian variety A over K such that
End0(A)
is Q-isomorphic to D (with the
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