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 (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
Cϕ where Cϕ 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 AΦ be the corresponding abelian variety over C. By
construction (and GAGA), we get an action by OL on AΦ and hence an embedding
L →
End0(AΦ)
as a subfield of Q-degree [L : Q] = 2 dim(AΦ). This makes AΦ 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
Cϕ
is the map (aϕ) → (ϕ(c)aϕ) involving multiplication in C. Thus, AΦ equipped with
the embedding L →
End0(AΦ)
gives rise to the CM type Φ on L.
The CM abelian varieties AΦ 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 AΦ vary
