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
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