6 INTRODUCTION

In Chapter 4 we show CM liftability after an isogeny over the finite ground field

(lifting over a characteristic zero local domain that need not be normal). That is,

every CM structure (A0,L →

End0(A0))

over a finite field κ has an

isogeny over κ to a CM structure (B0,L →

End0(B0))

that admits a

CM lifting;

(see 4.1.1). This statement is immediately reduced to the case when L is a CM

field (not just a CM algebra) and the whole ring OL of integers of L operates on

A0, which we assume.

Our motivation comes from the proof in 4.1.2 (using an algebraization argument

at the end of 4.1.3) that the counterexample in 2.3.1 to CM lifting over a normal

local domain satisfies this property. In general, after an easy reduction to the

isotypic case, we apply the Serre-Tate deformation theorem to localize the problem

at p-adic places v of the maximal totally real subfield L+ of a CM field L ⊆

End0(A0)

of degree 2 dim(A0). This reduces the existence of a CM lifting for the

abelian variety A0 to a corresponding problem for the CM p-divisible group A0[v∞]

attached to v.8

We formulate several properties of v with respect to the CM field L; any one

of them ensures the existence of a CM lifting of

A0[v∞]κ

after applying a κ-isogeny

to

A0[v∞]

(see 4.1.6, 4.1.7, and 4.5.7). These properties involve the ramification

and residue fields of L and

L+

relative to v. If v violates all of these properties

then we call it bad (with respect to

L/L+

and κ). Let Lv := L ⊗L+ Lv

+.

After

applying a preliminary κ-isogeny to arrange that OL ⊂ End(A0), for v that are

not bad we apply an OL-linear κ-isogeny to arrange that the Lie type of the OL,v-

factor of Lie(A0) (i.e., its class in a certain K-group of (OL,v/(p)) ⊗ κ-modules) is

“self-dual”. Under the self-duality condition (defined in 4.4.3) we produce an OL,v-

linear CM lifting of

A0[v∞]κ

by specializing a suitable OL,v-linear CM v-divisible

group in mixed characteristic; see 4.4.6. We use an argument with deformation

rings to eliminate the intervention of κ: if every p-adic place v of

L+

is not bad

then there exists a κ-isogeny A0 → B0 such that OL ⊂ End(B0) and the pair

(B0, OL → End(B0)) admits a lift to characteristic 0 without increasing κ.

If some p-adic place v of the totally real field

L+

is bad then the above argument

does not work because in that case no member of the

OL,v-linear κ-isogeny class of

the p-divisible group A0[v∞] has a self-dual Lie type. Instead we change A0[v∞] by a

suitable OL,v-linear κ-isogeny so that its Lie type becomes as symmetric as possible,

a condition whose precise formulation is called “striped”. Such a p-divisible group

is shown to be isomorphic to the Serre tensor construction applied to a special class

of 2-dimensional p-divisible groups of height 4 that are similar to the ones arising

from the abelian surface counterexamples in 2.3.1; we call these toy models (see

4.1.3, especially 4.1.3.2).

These “toy models” are suﬃciently special that we can analyze their CM lift-

ing properties directly; see 4.2.10 and 4.5.15(iii). After this key step we deduce

the existence of a CM lifting of

A0[v∞]κ

from corresponding statements for (the

p-divisible group version of) toy models. In the final step, once again we use de-

formation theory to produce an abelian variety B0 isogenous to (the original) A0

over κ and a CM lifting of B0 over a possibly non-normal 1-dimensional complete

local noetherian domain of characteristic 0 with residue field κ. Although OL acts

8See

1.4.5.3 for the statement of the Serre–Tate deformation theorem, and 2.2.3 and 4.6.3.1

for a precise statement of the algebraization criterion that is used in this localization step.