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.
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