went unnoticed for a long time. Once this distinction came in focus, answers to the
resulting questions became available.
If we ask for a CM lifting over a normal domain up to isogeny, in general a base
field extension before modification by an isogeny is necessary. This is explained in
2.1.2, where we formulate the “residual reflex obstruction”, the idea for which goes
as follows. Over an algebraically closed field K of characteristic zero, we know that
a simple CM abelian variety B with K-valued CM type Φ (for the action of a CM
field L) is defined over a number field in K containing the reflex field E(Φ) of Φ.
Suppose that for every K-valued CM type Φ of L, the residue field of E(Φ) at any
prime above p is not contained in the finite field κ with which we began in question
(a). In such cases, for every CM structure L →
on A0 and any abelian
variety B0 over κ which is κ-isogenous to A0, there is no L-linear CM lifting of
B0 over a normal local domain R of characteristic zero with residue field κ.6 In
2.3.1–2.3.3 we give such an example, a supersingular abelian surface A0 over Fp2
with End(A0) = Z[ζ5] for any p ≡ ±2 (mod 5). A much broader class of examples
is given in 2.3.5, consisting of absolutely simple abelian varieties (with arbitrarily
large dimension) over Fp for infinitely many p.
Note that passing to the normalization of a complete local noetherian domain
generally enlarges the residue field. Hence, if we drop the condition that the mixed
characteristic local domain R be normal then the obstruction in the preceding
consideration dissolves. And in fact we were put on the right track by mathematics
itself. The phenomenon is best illustrated in the example in 4.1.2, which is the
same as the example in 2.3.1 already mentioned: an abelian surface C0 over Fp2
with CM order Z[ζ5] that, even up to isogeny, is not CM liftable to a normal local
domain of characteristic zero. On the other hand, this abelian surface C0 is CM
liftable to an abelian scheme C over a mixed characteristic non-normal local domain
of characteristic zero, though the maximal subring of Z[ζ5] whose action lifts to C
is non-Dedekind locally at p; see
This example is easy to construct, and
the proof of the existence of a CM lifting, possibly after applying an Fp2 -rational
isogeny, is not diﬃcult either.
In Chapter 4 we show that the general question of existence of a CM lifting
after an appropriate isogeny can be reduced to the same question for (a mild gen-
eralization of) the example in 4.1.2, enabling us to prove:
every abelian variety A0 over a finite field κ admits an isogeny A0 → B0
over κ such that B0 admits a CM lifting to a mixed characteristic local
domain with residue field κ.
There are refined lifting problems, such as specifying at the beginning which CM
structure on A0 is to be lifted, or even what its CM type should be on a geometric
fiber in characteristic 0. These matters will also be addressed.
source of obstructions is that the base field κ might be too small to contain at least
one characteristic p residue field of the reflex field E(Φ) for at least one CM type Φ on L. Thus,
the field of definition of the generic fiber of the hypothetical lift may be too big. Likewise, an
obstruction for question (b) is that the field of definition of the p-divisible group
too big (in a sense that is made precise in 3.8.3 and illustrated in 3.8.4–3.8.5).
modification by isogeny is necessary in this example, but the universal deformation for
C0 with its Z[ζ5]-action is a non-algebraizable formal abelian scheme over W (Fp2 ).