Our basic method is to “localize” various CM lifting problems to the corre-
sponding problems for p-divisible groups. Although global properties of abelian va-
rieties are often lost in this localization process, the non-rigid nature of p-divisible
groups can be an advantage. In Chapter 3 the size of fields of definition of a p-
divisible group in characteristic p appears as an obstruction to the existence of CM
lifting. The reduction steps in Chapter 4 rely on a classification and descent of
CM p-divisible groups in characteristic p with the help of their Lie types (see 4.2.2,
4.4.2). In addition, the “Serre tensor construction” is applied to p-divisible groups,
both in characteristic p and in mixed characteristic (0,p); see 1.7.4 and 4.3.1 for
this general construction.
Survey of the contents. In Chapter 1 we start with a survey of general facts
about CM abelian varieties and their endomorphism algebras. In particular, we
discuss the deformation theory of abelian varieties and p-divisible groups, and we
review results in Honda-Tate theory that describe isogeny classes and endomor-
phism algebras of abelian varieties over a finite field in terms of Weil integers. We
conclude by formulating various CM lifting questions in 1.8. These are studied in
the following chapters. We will see that the questions can be answered with some
In Chapter 2 we formulate and study the “residual reflex condition”. Using this
condition we construct several examples of abelian varieties over finite fields κ such
that, even after applying a κ-isogeny, there is no CM lifting to a normal local
domain with characteristic zero and residue field of finite degree over κ; see 2.3. It
is remarkable that many such examples exist, but we do not know whether we have
characterized all possible examples; see 2.3.7.
We then study algebraic Hecke characters and review part of the theory of com-
plex multiplication due to Shimura and Taniyama. Using the relationship between
algebraic Hecke characters for a CM field L and CM abelian varieties with CM by
L (the precise statement of which we review and prove), we use global methods to
show that the residual reflex condition is the only obstruction to the existence of
CM lifting up to isogeny over a normal local domain of characteristic zero. We also
give another proof by local methods (such as p-adic Hodge theory).
In Chapter 3 we take up methods described in [93]. In that paper classical CM
theory in characteristic zero was used. Here we use p-divisible groups instead of
abelian varieties and show that the size of fields of definition of a p-divisible group
in characteristic p is a non-trivial obstruction to the existence of a CM lifting. In 3.3
we study the notion of isogeny for p-divisible groups over a base scheme (including
its relation with duality). We show, in one case of the CM lifting problem left
open in [93, Question C], that an isogeny is necessary. Our methods also provide
effectively computed examples. Some facts about CM p-divisible groups explained
in 3.7 are used in 3.8 to get an upper bound of a field of definition for the closed
fiber of a CM p-divisible group.
In Appendix 3.9, we use the construction (in 3.7) of a p-divisible group with
any given p-adic CM type over the reflex field to produce a semisimple abelian
crystalline p-adic representation of the local Galois group such that its restriction to
the inertia group is “algebraic” with algebraic part that we may prescribe arbitrarily
in accordance with some necessary conditions (see 3.9.4 and 3.9.8).
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