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.