1.5. CM TYPES 67

(without repetition) through all L-linear isogeny classes of (necessarily isotypic)

CM abelian varieties over C with complex multiplication by L.

1.5.3.1. Definition. Let A be an isotypic abelian variety of dimension g 0

over a field K, and let L be a CM field of degree 2g equipped with an embedding

j : L →

End0(A).

The dual CM structure on the dual abelian variety At is

the embedding L →

End0(At)

defined by x →

j(x)t,

where x → x is complex

conjugation on L.

It is easy to check that this definition respects double duality (i.e., if

Att

is

equipped with the CM structure dual to the one on

At

then the canonical isomor-

phism A

Att

is L-linear). The reason for the appearance of complex conjugation

on L in the definition of the dual CM structure is that when K is algebraically

closed of characteristic 0 it gives

At

the same (K-valued) CM type as A.

To verify this equality of CM types we may reduce to the case when K = C and

then use the exhaustive construction in the complex-analytic theory as in Exam-

ple 1.5.3. Alternatively, still working over C, consider the functorial isomorphism

Lie(At) H1(A, OA) and the functorial Hodge decomposition

C ⊗Q H1(A(C), Q)

H1(A(C), C)∨

Lie(A) ⊕

H1(A, OA)∨.

Since H1(A(C), Q) is 1-dimensional as an L-vector space, when

At

is equipped with

the dual action

j(x)t

(without the intervention of complex conjugation) then its

CM type is Hom(L, C) − Φ = Φ.

1.5.4. Descent to a number field. For a CM abelian variety over an alge-

braically closed field K of characteristic 0, we may make the CM type essentially

be independent of K by replacing K with Q (see Remark 1.5.2.2). This enables us

to use the complex-analytic theory to prove the following purely algebraic result.

1.5.4.1. Proposition. Let K be an algebraically closed field of characteristic 0.

Let L be a CM field, and consider a CM abelian variety A over K with complex

multiplication via j : L →

End0(A).

The L-linear isogeny class of A is uniquely

determined by the K-valued CM type Φ on L associated to (A, j), and every CM

type on L arises in this way from some (A, j) over K.

The hypothesis K = K cannot be weakened. For example, if K is a number

field containing a Galois closure of L over Q (so all K-valued CM types on L

are K-valued) then any quadratic twist of A (equipped with the evident L-linear

structure) has the same CM type as (A, j) but is generally not K-isogenous to A.

Proof. In view of Lemma 1.2.1.2, by expressing K as a direct limit of alge-

braically closed subfields of finite transcendence degree over Q we can reduce to the

case when K has finite transcendence degree over Q. To show that the CM type

determines the L-linear isogeny class it suﬃces (again by Lemma 1.2.1.2) to treat

the case K = C. This case was addressed in Example 1.5.3 via the complex-analytic

theory, where it was also seen that every CM type Φ on L does arise when K = C.

It remains to show that every CM type Φ on L arises when K = Q. Consider

the CM abelian variety AΦ over C with complex multiplication by L and CM type

Φ as in Example 1.5.3. Recall that OL = L ∩ End(AΦ) inside

End0(AΦ).

By

expressing C as a direct limit of its finitely generated Q-subalgebras, there is such a