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.
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