INTRODUCTION 7

on the closed fiber, we can only ensure that a subring of OL of finite

index9

acts

on the lifted abelian scheme (see 4.6.4).

Appendix A. In Appendix A.1 we provide a self-contained development of the

proof of the p-part of Tate’s isogeny theorem for abelian varieties over finite fields of

characteristic p, as well as a proof of Tate’s formula for the local invariants at p-adic

places for endomorphism algebras of simple abelian varieties over such fields. (An

exposition of these results is also given in [79]; our treatment uses less input from

non-commutative algebra.) Appendices A.2 and A.3 provide purely algebraic proofs

of the Main Theorem of Complex Multiplication for abelian varieties, as well as a

converse result, both of which are used in essential ways in Chapter 2. In Appendix

A.4 we use Shimura’s method to show that an algebraic Hecke character with a

given algebraic part can be constructed over the field of moduli of the algebraic

part, with control over places of bad reduction.

In the special case of the reflex norm of a CM type (L, Φ), combining this

construction of algebraic Hecke characters with the converse to the Main Theorem

of CM in A.3 proves that over the associated field of moduli M ⊂ Q (a subfield of

the Hilbert class field of the reflex field E(L, Φ)) there exists a CM abelian variety

A with CM type (L, Φ) such that A has good reduction at all p-adic places of

M; see A.4.6.5. Since M is the smallest possible field of definition given (L, Φ),

this existence result is optimal in terms of its field of definition. Typically M =

E(L, Φ), and this is regarded as a “class group obstruction” to finding A with its

CM structure by L over E(L, Φ), a well-known phenomenon in the classical CM

theory of elliptic curves.

(In the “local” setting of CM p-divisible groups over p-adic integer rings there

are no class group problems and one gets a better result: in 3.7 we use the preceding

global construction over the field of moduli to prove that for any p-adic CM type

(F, Φ) and the associated p-adic reflex field E ⊂ Qp over Qp there exists a CM

p-divisible group over OE with p-adic CM type (F, Φ).)

Appendix B. In Appendices B.1 and B.2, we give two versions of a more di-

rect (but more complicated) proof of the existence of CM liftings for a higher-

dimensional generalization of the toy

model.10

The first version uses Raynaud’s

theory of group schemes of type (p,...,p). The second version uses recent devel-

opments in p-adic Hodge theory. We hope that material described there will be

useful in the future. In Appendix B.3 we compare several Dieudonn´ e theories over

a perfect base field of characteristic p 0. In Appendix B.4 we give a formula for

the Dieudonn´ e module of the closed fiber of a finite flat commutative group scheme,

constructed using integral p-adic Hodge theory; this formula is used in B.2.

9This

subring of finite index can be taken to be Z + pOL.

10In

the original proof of our main CM lifting result in 4.1.1, the case of a bad place v|p of

L+

was reduced through the Serre tensor construction to this existence result. Both B.1 and B.2

are logically independent of results in Chapter 4. Readers who cannot wait to see a proof of the

existence of a CM lifting (without modification by any isogeny) for such a higher-dimensional toy

model may proceed directly to B.1 or B.2, after consulting 4.2 for the definition of the Lie type

of an O-linear p-divisible group and related notation.