30 1. ALGEBRAIC THEORY OF COMPLEX MULTIPLICATION
Moreover, by [82, §21, Thm. 2] it can be arranged that under this composite
isomorphism the positive involution on D goes over to transpose on each factor
Mat2(Zv) = Mat2(R). Thus, for D of Type II the fixed part of the involution
on D has Q-dimension 2e and hence Z-degree 2. By centrality of Z in the
division algebra D, the condition
x∗
= x for x in D of Type II therefore
defines a necessarily commutative quadratic extension Z of Z inside D, so g
is divisible by [Z : Q] = 2e as desired.
The preceding results are summarized in the following table, taken from the
end of [82, §21]. (As we have just seen, the hypothesis there that K is algebraically
closed is not necessary.) The invariants of D =
End0(A)
are given in the first three
columns. In the last two columns we give some necessary divisibility restrictions
on these invariants.
Type e d char(K) = 0 char(K) 0
I e = e0 1 e | g e | g
II e = e0 2 2e | g 2e | g
III e = e0 2 2e | g e | g
IV e = 2e0 d
e0d2
| g e0d | g
We refer the reader to [82, §21], and to [92] for further information on these
invariants. Using the above table, we can prove the following additional facts when
the simple A admits sufficiently many complex multiplications.
1.3.6.4. Proposition. Let A be a simple abelian variety of dimension g 0 over
a field K, and assume that A admits sufficiently many complex multiplications. Let
D =
End0(A).
(1) If char(K) = 0 then D is of Type IV with d = 1 and e = 2g (so D is a CM
field, by Theorem 1.3.6.2).
(2) If char(K) 0 then D is of Type III or Type IV.
Proof. By simplicity, D is a division algebra. Its center Z is a commutative field.
First suppose char(K) = 0. Let P D be a commutative semisimple Q-
subalgebra with [P : Q] = 2g. Since D is a division algebra, P is a field. The above
table (or the discussion preceding it) says that the degree [D : Q] = ed2 divides
[P : Q] = 2g, so the inclusion P D is an equality. Thus, D is commutative
(i.e., d = 1), so D = Z is a commutative field and hence e := [Z : Q] = 2g by
the complex multiplication hypothesis. The table shows that in characteristic 0 we
have e|g for Types I, II, and III, so D is of Type IV.
Suppose char(K) 0. In view of the divisibility relations in the table in positive
characteristic, D is not of Type I since in such cases D is a commutative field whose
Q-degree divides dim(A), contradicting the existence of sufficiently many complex
multiplications. For Type II we have 2e|g, yet the integer 2e = 2[Z : Q] is the Q-
degree of a maximal commutative subfield of the central quaternion division algebra
D over Z, so there are no such subfields with Q-degree 2g. Since a commutative
semisimple Q-subalgebra of D is a field (as D is a division algebra), Type II is not
possible if the simple A has sufficiently many complex multiplications.
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