INDEX 387
height of, 39
isoclinic, 139
isogeny of, 148–152
local-local, 140
local-local part, 140
of multiplicative type, 140
ordinary, 140
quasi-polarization of, 60
Serre dual of, 39
slopes of, see slopes, of a p-divisible
group
with sufficiently many complex
multiplication, 169
polarization, 37
reciprocity laws
sign conventions of, 10
reduction map from p-adic CM types to Lie
types, 212, 230
reflex field
of a CM type, 93
of a p-adic cocharacter, 187
of a p-adic CM type, 170, 172, 182, 183
reflex norm
of a CM type, 94, 95
of a p-adic cocharacter, 187, 189
of a p-adic CM type, 171
of a p-adic cocharacter, 188
replete divisor, 308, 309
degree of, 308
principle, 308
residual reflex condition, 100
necessary and sufficient condition for
(IN), 100, 128–136
self-dual Lie type, 230, 232, 235, 236, 239,
242, 242
self-dual p-adic CM type, 206, 208, 209,
230, 232, 235, 238, 239, 242
self-dual up to isogeny, 243, 244
self-duality
and algebraization, 243
and reduction modulo p, 231
condition, 199, 206, 208, 209, 228, 230,
243, 244, 248
Serre group attached to a number field, 120
neutral component, 119, 299
character group of, 299
weight cocharacter of, 303
Serre tensor product construction
and Lie types, 227
for abelian varieties, 80, 83, 224
for p-divisible groups, 238
for p-divisible groups, 224, 236
Serre-Tate canonical lifting, 60
Serre-Tate theorem on deformation of
abelian varieties, 59, 206
Shimura–Taniyama formula, 98
and (IN), 100
and residual reflex condition, 100
for a CM abelian variety, 98
for an algebraic Hecke character, 313
short exact sequence, 34
singular j-invariant, 1
Skolem–Noether theorem, 18
slopes
of a Lie type, 214, 217, 233, 233
of a p-divisible group, 139, 139, 207, 215,
237, 322
of an abelian variety, 98
striped Lie type, see Lie type, striped
supersingular j-values, 1
Tate’s theorem
and (RIN), 75
CM structure for abelian varieties with
sufficiently many complex
multiplication, 26
extending homomorphisms between
p-divisible groups, 58
Hodge-Tate decomposition for p-divisible
groups, 58
homomorphisms between abelian
varieties over a finite field, 70
toy model, 110
CM abelian surface, 196, 203
classification of, 220
CM p-divisible group, 204, 209, 236–239
higher dimensional, 321, 333
truncated Barsotti-Tate group
BT1 group scheme, 144
BTn group, 143
uniform Lie type, 233, 235, 242, 242, 244
Weil q-integer, 70
Weil q-number
of weight w, 70, 304, 313, 315
existence, 318
slopes of, 315
Weil restriction of scalars, 22
windows, 362
Witt covectors, 345, 345–347
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