386 INDEX

to normal domains up to isogeny after

finite residual field extension (RIN)

Honda-Tate theorem on (RIN), 75

to normal domains up to isogeny after

finite residue field extension (RIN), 88

up to isogeny (I), 88

existence, 195

CM order, 65, 68, 80, 89

CM p-divisible group, 167

existence, 172, 178

Galois representation of, 176

uniqueness up to isogeny, 173

CM structure

dual, 67, see also CMtype of the dual of

a CM abelian variety

for abelian varieties, 2, 32

CM type

for a CM algebra, 66

for a CM field, 66

of a CM abelian variety, 66

determines the isogeny class, 67

of the dual of a CM abelian variety, 67

p-adic, see p-adic CM type

valued in a field, 66

co-Lie complex, 356

counterexample

to (CML) and (R), 183–184

to (IN), 101, 110

with two slopes, 111–114

to CM lifting with action by the full ring

of integers in the CM field, 198

crystalline Dieudonn´ e theory, 354–359

deformation ring

for a CM structure, 202

of a CM structure, 92

of a p-divisible groups, 169

deformation ring argument, 91, 106, 169,

233

deformation theory

for abelian schemes, 196

for p-divisible groups, 196

Dieudonn´ e theory, 38, 137–138, 347–351

basic differential invariants, 356–359

comparison of Dieudonn´ e theories, 356

Dieudonn´ e ring, 38

Dieudonn´ e-Manin classification, 139

dimension of a Lie type, 212

display, 354, 362–368

duality theorem

for abelian varieties, 37

for p-divisible groups, 152

effective elements

in Rk(OF ) and Rκ(OF ), 211, 212, 229

extended Lubin-Tate type, 156

field of definition

of a p-adic cocharacter, 187

of a cocharacter, 315

field of definition as obstruction to CML,

181–185

field of moduli

of an admissible algebraic

homomorphism, 310

good

p-adic place, 208, 208, 209, 247

pair, 209, 235, 238

Grothendieck group, 207, 208, 211, 212,

227, 241, 324

Hodge-Tate decomposition, 168, 185, 189

Honda-Tate theorem, 71

and CM lifting, 3

Kisin modules, 360–362

level structure

finite ´ etale, 13, 57

Lie type, 212, 229

and Galois descent, 207, 211, 217

rational over a field, 215, 236, 237, 242

self-dual, 230, 235, 236, 239, 242, 242

striped, 208, 234, 234, 236, 238, 239

local deformation space

for a CM structure, see deformation ring,

for a CM structure

of a p-divisible group, 169

main theorem of complex multiplication,

127, 257–292

algebraic form, 266

analytic form, 288

converse to, 128, 292–296

multiplicities of a Lie type, 233

Newton polygon, 114

and (IN), 114–116

order lattice, 286

p-adic abelian crystalline representation,

174, 189

algebraic on the inertia subgroup, 190

p-adic CM type, 168

compatible with a given CM structure,

170

of a CM p-divisible group, 168, 206

self-dual, 206, 208, 209, 230, 232, 235,

238, 239, 242

p-adic Hodge theory, 333, 359

p-divisible group, 39

a-number of, 143

and deformation of abelian varieties, 59

connected, 40, 41

Dieudonn´ e-Manin classification up to

isogeny, 139

´ etale, 40, 140