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.
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