66 1. ALGEBRAIC THEORY OF COMPLEX MULTIPLICATION
K with K. We may then reduce to the case K = K = C, in which case a proof is
given via the complex-analytic uniformization in [82, §22].
The preceding considerations motivate the following concepts.
1.5.2.1. Definition. (i) Let L be a CM field of degree 2g over Q and K an
algebraically closed field of characteristic 0. An K-valued CM type for L is a subset
Φ Hom(L, K) of representatives for the g orbits of the action by the complex
conjugation ι of L. That is, Φ consists of g distinct elements such that ϕ ι Φ
for all ϕ Φ, or equivalently Hom(L, K) = Φ ι). To emphasize the role of
L, we often refer to the pair (L, Φ) as a CM type.
(ii) Let L = L1 × · · · × Ls be a CM algebra, where each Li is a CM field. A
CM type for L is a subset Φi Hom(Li,K) = Hom(L, K) where (Li, Φi) is a
CM type for each i.
If K is a field of characteristic 0 and K /K is an algebraically closed extension,
then the tangent space to a CM abelian variety A over K with complex multipli-
cation by a CM algebra L determines a K -valued CM type Φ for L (apply Lemma
1.5.2 to the isogeny factors of A determined by the primitive idempotents of L).
This is an invariant of the L-linear isogeny class of A over K.
1.5.2.2. Remark. In general, a CM type takes values in the algebraic closure Q
of Q in K , so if we first choose this algebraic closure as an abstract field and then
take K
to be equipped with a specified embedding of this Q then we can regard the
CM type as being independent of K ; this is sometimes useful for passing between
different choices of K (such as C and Qp).
1.5.3. Example. Let L be a CM field and Φ a C-valued CM type on L. Let
(R ⊗Q L)Φ denote R ⊗Q L =
v|∞
Lv endowed with the complex structure defined
via the isomorphism Lv C using the unique ϕv Φ pulling back the standard
absolute value of C to the place v of L for each v|∞. In other words, (R ⊗Q L)Φ =
ϕ∈Φ
where denotes C equipped with the L-action via ϕ : L C. The ring
of integers OL is a lattice in R⊗Q L = R⊗Z OL in the natural way, so (R⊗Q L)Φ/OL
is a complex torus of dimension [L : Q]/2.
In the complex-analytic theory [82, §22] it is proved (using that L is a CM
field) that this complex torus admits a Riemann form (with respect to which the
action of each c L has adjoint given by the complex conjugate c L), and hence
is an abelian variety. Let be the corresponding abelian variety over C. By
construction (and GAGA), we get an action by OL on and hence an embedding
L
End0(AΦ)
as a subfield of Q-degree [L : Q] = 2 dim(AΦ). This makes into
a CM abelian variety over C with complex multiplication by L. The action by any
c OL End(AΦ) on
Lie(AΦ) = Lie(AΦ
an)
= (R ⊗Q L)Φ =
ϕ∈Φ

is the map (aϕ) (ϕ(c)aϕ) involving multiplication in C. Thus, equipped with
the embedding L
End0(AΦ)
gives rise to the CM type Φ on L.
The CM abelian varieties are generally not simple; see Remark 1.5.4.2 for
further discussion of the simplicity aspect. It is shown in the classical theory [82,
§22, First Ex., Thm.] that as we vary Φ through all CM types on L, the vary
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