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

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