32 1. ALGEBRAIC THEORY OF COMPLEX MULTIPLICATION

Proof. By Theorem 1.3.4, the endomorphism algebra

End0(A)

contains a com-

mutative field with Q-degree 2 dim(A). This property is preserved after any ground

field extension (even though the endomorphism algebra may get larger), so by the

final part of Theorem 1.3.1.1 isotypicity is preserved as well.

1.3.8. CM abelian varieties. It turns out to be convenient to view the CM

algebra P in Theorem 1.3.4 as an abstract ring in its own right, and to thereby

regard the embedding P →

End0(A)

as additional structure on A. This is encoded

in the following concept.

1.3.8.1. Definition. Let A be an abelian variety over a field K, and assume that

A has suﬃciently many complex multiplications. Let j : P →

End0(A)

be an

embedding of a CM algebra P with [P : Q] = 2 dim(A). Such a pair (A, j) is called

a CM abelian variety (with complex multiplication by P ).

Note that in this definition we are requiring P to be embedded in the endo-

morphism algebra of A over K (and not merely in the endomorphism algebra after

an extension of K). For example, according to this definition, no elliptic curve over

Q admits a CM structure (even if such a structure exists after an extension of the

base field).

As an application of Theorem 1.3.4, we establish the following result concerning

the possibilities for Z of Type III in Proposition 1.3.6.4(2). This will not be used

later.

1.3.8.2. Proposition. Let A, K, and D be as in Proposition 1.3.6.4(2) with p =

char(K) 0, and let Z be the center of D, g = dim(A), d = [D : Z], and

e = [Z : Q]. We have ed = 2g, and if D is of Type III (so d = 2) then either Z = Q

or Z = Q(

√

p).

Note that in this proposition, K is an arbitrary field with char(K) 0; K is

not assumed to be finite.

Proof. We always have ed|2g, but ed = [D : Q] and D contains a field P of

Q-degree 2g, so 2g|ed. Thus, ed = 2g. Now we can assume A is of Type III, so the

field Z is totally real.

Since A is of finite type over K and D is finite-dimensional over Q, by direct

limit considerations we can descend to the case when K is finitely generated over

Fp. Let S be a separated integral Fp-scheme of finite type whose function field is

K. Since A is an abelian variety over the direct limit K of the coordinate rings

of the non-empty aﬃne open subschemes of S, by replacing S with a suﬃciently

small non-empty aﬃne open subscheme we can arrange that A is the generic fiber

of an abelian scheme A → S. Since S is connected, the fibers of the map A → S

all have the same dimension, and this common dimension is g (as we may compute

using the generic fiber A).

The Z-module End(A) is finitely generated, and each endomorphism of A ex-

tends uniquely to a U-endomorphism of AU for some non-empty open U in S

(with U perhaps depending on the chosen endomorphism). Using a finite set of

endomorphisms that spans End(A) allows us to shrink S so that all elements of

End(A) extend to S-endomorphisms of A , or in other words End(A) = End(A ).

We therefore have a specialization map D =

End0(A)

→

End0(As)

for every s ∈ S.