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