(1) Abelian varieties. In Mumford’s book [82] the theory of abelian varieties
is developed over an algebraically closed base field, and we need the theory
over a general field; references addressing this extra generality are Milne’s
article [76] (which rests on [82]) and the forthcoming book [45]. Since [45]
is not yet in final form we do not refer to it in the main text, but the reader
should keep in mind that many results for which we refer to [82] and [76]
are also treated in [45]. We refer the reader to [83, Ch. 6, §1–§2] for a self-
contained development of the elementary properties of abelian schemes, which
we freely use. (For example, the group law is necessarily commutative and is
determined by the identity section, as in the theory over a field.)
(2) Semisimple algebras. We assume familiarity with the classical theory of
finite-dimensional semisimple algebras over fields (including the theory of their
splitting fields and maximal commutative subfields). A suitable reference for
this material is [53, §4.1–4.6]; another reference is [11]. In 1.2.2–1.2.4 we
review some of the facts that we need from that theory.
(3) Descent theory and formal schemes. In many places, we need to use the
techniques of descent theory and Grothendieck topologies (especially the fppf
topology, though in some situations we use the fpqc topology to perform de-
scent from a completion). This is required for arguments with group schemes,
even over a field, such as in considerations with short exact sequences. For
accounts of descent theory, we refer the reader to [10, §6.1–6.2], and to [39,
Part 1] for a more exhaustive discussion. These techniques are discussed in a
manner well-suited to group schemes in [98] and [30, Exp. IV–VIA].
Our arguments with deformation theory rest on the theory of formal
schemes, especially Grothendieck’s formal GAGA and algebraization theo-
rems. A succinct overview of these matters is given in [39, Part 4], and the
original references [34, I, §10; III1, §4–§5] are also highly recommended.
e theory and p-divisible groups. To handle p-torsion phenom-
ena in characteristic p 0 we use Dieudonn´ e theory and p-divisible groups.
Brief surveys of some basic definitions and properties in this direction are
given in 1.4, 3.1.2–3.1.6, and B.3.5.1–B.3.5.5. We refer the reader to [119],
[71] and [110, §6] for more systematic discussions of basic facts concerning
p-divisible groups, and to [29] and [41, Ch. II–III] for self-contained devel-
opments of (contravariant) Dieudonn´ e theory, with applications to p-divisible
groups. Contravariant Dieudonn´ e theory is used in Chapters 1–4.
Covariant Dieudonn´ e theory is used in Appendix B.1 because the alter-
native proof there of the main result of Chapter 4 uses a covariant version of
p-adic Hodge theory. A brief summary of covariant Dieudonn´ e can be found
in B.3.5.6–B.3.6.7. We recommend [136] for Cartier theory; an older standard
reference is [69].
A very useful technique within the deformation theory of p-divisible groups
is Grothendieck–Messing theory, which is developed from scratch in [75]. Al-
though we do not provide an introduction to this topic, we hope that our
applications of it may inspire an interested reader who is not familiar with
this technique to learn more about it.
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