90 1. ALGEBRAIC THEORY OF COMPLEX MULTIPLICATION

CM lifting of (B, L ⊂

End0(B))

over the valuation ring of a subfield of Qp of finite

degree over W (Fq)[1/p], the associated Qp-valued CM type Φ on L must satisfy the

following compatibility condition with the L-slopes:

#{φ ∈ Φ | φ induces w on L} = rw

for all w; see 2.1.4.1–2.1.4.2. For any Qp-valued CM type (L, Φ) satisfying this

necessary condition, in Chapter 2 we prove a necessary and suﬃcient condition for

the existence of a solution B to problem (IN) for (B, L) over a local normal domain

R ⊇ W (Fq) with residue field Fq such that the L-action (in the isogeny category)

lifts to B and the resulting CM structure over K = Frac(R) has Qp-valued CM

type Φ relative to some W (Fq)[1/p]-algebra embedding of Qp into K.

Without normality it is more diﬃcult to determine which CM types on L arise

from solutions to the lifting problem (I) for (B, L) when we require the L-action to

lift. If L is a CM field, the answer is p-local for the maximal totally real subfield

L+: it can be analyzed separately for each p-adic place v of L+. To be precise, a

Qp-valued CM type Φ for L corresponds to a sequence {(Lv + ⊗L+L, Φv)}v where Φv

is a set of Qp-algebra homomorphisms Lv

+

⊗L+ L → Qp, and via p-divisible groups

the question reduces to determining (for each v) the family Fv of all Φv that arise

as the v-component of the Qp-valued CM type of an aﬃrmative solution to the CM

lifting problem (I) for the CM structure (B, L ⊂

End0(B)).

When v is split in L, say with w and w the two places over it on L, there turn

out to be no restrictions on Φv beyond the above compatibility conditions

rw = #{φ ∈ Φv | φ induces w on L}, rw = #{φ ∈ Φv | φ induces w on L}

(which can always be satisfied, since any φ ∈ Φv induces w or w on L). However

when v is non-split in L, the proofs in Chapter 4, B.1, and B.2 provide only a

non-empty subset Fv of

Fv;3

we do not know the discrepancy between Fv and

Fv.4

3The

set Fv is the family of all v-components of Qp-valued CM types of aﬃrmative solutions

to (I) which can be constructed by the method in B.1 and B.2 together with the Serre tensor

construction for p-divisible groups in 4.3. The proof in Chapter 4, especially 4.5.15 (iii)–4.5.17,

which uses the Serre tensor construction and the existence of CM lifting of toy model p-divisible

groups, gives a non-empty subset Fv of Fv which can be strictly smaller than Fv.

4A

complete solution of (sCML) should also provide an answer to this question.