1.3. COMPLEX MULTIPLICATION 29

given involution on D arising from a polarization on A); see [1], [2], [3], [112, §4.1,

Thm. 5], and [46, Thm. 13]. For a survey and further references on this topic, see

[92]. We will not need this result.

Instead, we are interested in the non-trivial constraints on the Albert algebras

that arise from polarized simple abelian varieties A over an arbitrary field K when

char(K) and dim A are fixed. Before listing these constraints, it is convenient to

record Albert’s classification of general Albert algebras (omitting a description of

the possibilities for the involution).

1.3.6.2. Theorem (Albert). Let (D, (·)∗) be an Albert algebra. For any place v

of the center Z, let v∗ denote the pullback of v along x → x∗. Exactly one of the

following occurs:

Type I: D = Z is a totally real field.

Type II: D is a central quaternion division algebra over a totally real field Z

such that D splits at each real place of Z.

Type III: D is a central quaternion division algebra over a totally real field Z

such that D is non-split at each real place of Z.

Type IV: D is a central division algebra over a CM field Z such that for all

finite places v of Z, invv(D) + invv∗ (D) = 0 in Q/Z and moreover D splits at such

a v if v =

v∗.

Proof. See [82, §21, Thm. 2] (which also records the possibilities for the involu-

tion).

1.3.6.3. Let A be a simple abelian variety over a field K, D =

End0(A),

and Z the

center of D. Let

Z+

be the maximal totally real subfield of Z, so either Z =

Z+

or Z is a totally complex quadratic extension of

Z+.

The invariants e = [Z : Q],

e0 = [Z+ : Q], d = [D : Z], and g = dim(A) satisfy some divisibility restrictions:

• whenever char(K) = 0, the integer

ed2

= [D : Q] divides 2g (proof: there

is a subfield K0 ⊆ K finitely generated over Q such that A descends to an

abelian variety A0 over K0 and the D-action on A in the isogeny category

over K descends to an action on A0 in the isogeny category over K0, so upon

choosing an embedding K0 → C we get a Q-linear action by the division

algebra D on the 2g-dimensional homology H1(A0(C), Q)),

• the action of D on V (A) with = char(K) implies (via Cor. to Thm. 4 of [82,

§19], whose proof is valid over any base field) that ed|2g,

• the structure of symmetric elements in

Q ⊗Z Hom(A,

At)

Q ⊗Z

Pic(A)/Pic0(A)

(via [82, §20, Cor. to Thm. 3], whose proof is valid over any base field) yields

that [L : Q]|g for every commutative subfield L ⊂ D whose elements are

invariant under the involution.

• for Type II in any characteristic we have 2e|g (which coincides with the general

divisibility

ed2|2g

when char(K) = 0 since d = 2 for Type II). To prove it

uniformly across all characteristics, first note that for Type II we have

R ⊗Q D = (R ⊗Q Z) ⊗Z D =

v|∞

Zv ⊗Z D

Mat2(Zv)e.