Index

a-number, 143

abelian scheme, 14, 46, 47

dual, 36

isogeny, 37

polarization, 37

abelian variety

absolutely simple, 15

and primary extension of fields, 15

isotypic, 22, 23, 31

lifting to characteristic 0, 86

Poincar´ e reducibility, 16

simple over a field, 15

abelian variety with suﬃciently many

complex multiplication, 24, 25, 77

Grothendieck’s theorem in char. p 0,

79

refinement, 86

admissible algebraic homomorphism, 301

and Weil numbers, 315

construction of, 316

field of moduli, 310

primitive, 299

weight of, 303

Albert algebra, 28

classification of, 29

algebraic Hecke character, 118

algebraic part of, 119

and complex multiplication, 127, 128

compatible system of -adic characters

attached to, 121

construction by surgery procedure, 124

equivalent definition, 120

existence over field of moduli, 311

existence theorem, 305

formula for, see Shimura Taniyama

formula, for an algebraic Hecke

character

weight of, 300

α-group scheme, 142

Dieudonn´ e module of, 358–359

bad

p-adic place, 208, 208, 209, 210, 246

pair, 209, 235, 236, 238

Barsotti-Tate group, see also p-divisible

group

truncated, see also truncated

Barsotti-Tate group

truncated at level 1, 144

Brauer group, 18

of a global field, 20

order of an element of, 20

of a local field, 19

Cartier dual, 34, 326

Cartier theory, 351–353

Cartier module, 322, 352

V -flat, 352

V -reduced, 352

Cartier ring, 351

and the smooth formal group W , 351

central simple algebra, 16

Wedderburn’s theorem, 17

CM abelian variety, 23, 32

descent to a number field, 67

existence over field of moduli, 311

isotypic, 31

L-function of, 272

potential good reduction, 78

CM algebra, 26

CM field, 26

CM formal abelian scheme

criterion for algebraicity, 106

CM lifting

of a p-divisible group, 169

CM lifting questions, 86

after finite residue field extension (R), 88

CM lifting (CML), 87

suﬃcient condition, 102

Lie type of the closed fibers of CM

abelian schemes isogenous to a CM lift

(LTI), 210

strong CM lifting (sCML), 88, 210

to normal domains up to isogeny (IN), 88

local method, 178

necessary and suﬃcient condition, 100,

128–136

385