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