1.8. CM LIFTING QUESTIONS 89

to a CM structure over R. We will give examples to show that the choice of L can

affect the the answer to some of the lifting questions. Even if we know for a given

B and L ⊂

End0(B)

that there is a CM lift to characteristic 0 on which the action

of an order in L also lifts, it could be that the CM order L ∩ End(B) does not

lift. We will give examples where this happens in 4.1.2 (see Theorem 4.1.1 and the

non-algebraizable universal formal deformation in 4.1.2.3).

1.8.6. Answers to CM lifting problems. The proofs of the following answers

form the backbone of subsequent chapters.

1. By [93, Thm. B], for any g 2 there exist g-dimensional abelian varieties

over an algebraic closure of Fp for which there is no CM lift to characteristic

0. These results are proved in a much stronger form in Chapter 3. Thus,

(R) does not hold in general, so in particular (CML) sometimes fails to hold.

Hence, an isogeny is necessary; that is, it is better to consider (I) than (CML).

2. Building on lifting results for p-divisible groups in Chapter 3, in Chapter 4 we

prove that (I) holds for every B (so a strengthening of (RIN) holds, applying

the isogeny before making a finite extension on the residue field). In fact,

for any CM maximal commutative subfield L ⊂

End0(B)

we construct an

isogeny B → B to an abelian variety over Fq such that B has a CM lift to

characteristic 0 on which the action of the order Z + pOL in OL also lifts.

The abelian variety B generally depends on L, but we can arrange that

the isogeny to B is a p-power at most

p4g2

(and examples show that we cannot

arrange it to have degree not divisible by p in general). The CM lifting of B

can be found over an order in a p-adic field whose relative ramification degree

over a specific p-adic reflex field is tightly controlled (usually 1).

3. In contrast with success for (I), if we require R to be normal and do not

increase the residue field (but permit isogenies) then the answer is negative:

in Chapter 2 we give examples for which (IN) fails. Hence, for the existence

of a CM lifting to a normal domain of characteristic 0 we must allow a finite

extension of the initial finite field (and an isogeny), as in Theorem 1.6.5.1.

However, there is a salvage: for each B and choice of L ⊂

End0(B)

we

will give (in Chapter 2) concrete necessary and suﬃcient conditions in terms

of a Qp-valued CM type Φ on L for (IN) to have an aﬃrmative answer using

a CM lifting to which the action of an order in L (in the isogeny category)

also lifts and yields the specified CM type Φ. In Example 2.1.7 we will give B

for which this necessary and suﬃcient condition is satisfied for one choice of

L ⊂

End0(B)

(and a suitable Φ) but fails for another choice (and any Φ).

We expect that (sCML) in 1.8.5 does not hold in general, but this is a guess.

1.8.7. Remark. (CM types in (IN) and (I)). In the context of 1.8.5, for any CM

structure L ⊂

End0(B)

on the abelian variety B the q-Frobenius endomorphism

ϕ ∈ End(B) lies in

L×

since L is its own centralizer in

End0(B).

The combinatorial

data of L-slopes of the L-linear isogeny class of (B, L ⊂

End0(B))

is the sequence of

non-negative rational numbers rw = ordw(ϕ)/ordw(q) indexed by the set of places

w of L above p. When L is a CM field, with complex conjugation ι, this satisfies

the self-duality condition rw + rw◦ι = 1 for each w (since ϕ · ι(ϕ) = q in L).

Fix an algebraic closure Qp of Qp and an embedding of Fq into the residue field

of the valuation ring of Qp (so W (Fq)[1/p] canonically embeds into Qp). For any