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 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
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