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FGA A. Grothendieck, Fondements de Geometrie Algebrique, Seminaire Bourbaki, Exp. 190, 195, 212,221, 232, 236, ???

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MAV D. Mumford, Abelian Varieties, Oxford University Press, Oxford, 1970.

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Index

2-torsionof an elliptic curve, 81

3-torsionof an elliptic curve, 82

Abel’s Theorem, 223–224abelian varieties up to isogeny

category of, 181abelian variety

definition, 6over C, 75–76

arithmetic Frobenius, 310

Brauer equivalence, 309Brauer group, 309

canonical involution, 190, 311Cartier duality

of Frobenius and Verschiebung, 79central simple algebra, 309characteristic polynomial of Frobenius, 266complex abelian variety, 75–76complex torus, 75conjugate

q-Weil numbers, 289crystal, 246–262

definition of, 247e!ective, 247isogeny, 247

crystalline cohomology, 251cyclic algebra, 310

degreeof a central simple algebra, 309of a divisor, 8of a line bundle, 219

divisibilityof X(k), 74

divisordegree, 8e!ective, 8linear equivalence, 8on a curve, 8principal, 8

divisor class group, 8

e!ective!a-F -crystal, 247!a-F -isocrystal, 248

e!ective divisor, 8e!ective relative Cartier divisor, see relative Cartier

divisorelliptic curve, 9even theta characteristic, 228exponential map, 75

F -crystal, see crystal

F -isocrystal, see isocrystal

fixed point scheme, 71

flex point, 82

Frobenius, 76–80arithmetic, 310characteristic polynomial of, 266geometric, 265isogeny, 76iterated, 80

G-torsor, 174

Gauss map, 221

geometric Frobenius, 265

geometric line bundle, 1

geometric vector bundle, 1

groupdefinition, 5

group varietydefinition, 6

Hard Lefschetz Theorem, 271

Hasse-Weil bound, 273

Hasse-Weil Theorem, 266

Hasse-Weil-Serre bound, 273

heightof a !a-F -crystal, 248

hermitian form, 312

Hodge Index Theorem, 271

Hodge numbersof a crystal, 250

homomorphismdefinition, 13of abelian varieties, 12–14

indexof a central simple algebra, 309

involution, 311canonical, 190, 311of the first kind, 311of the second kind, 311positive, 313

irreduciblemodule, 308

isoclinic, 254

isocrystal, 246–262definition of, 247e!ective, 248

isogenous abelian varieties, 75, 80

– 321 –

isogeny, 72–80definition, 72degree, 72, 73equivalence relation, 75Frobenius, 76multiplication by n, 74of !a-F -crystals, 247purely inseparable, 73–74quasi-, 181separable, 73–74Verschiebung, 80

iterated Frobenius, 80, 265iterated Verschiebung, 80

Jacobi’s Inversion Theorem, 224Jacobian

definition of, 91, 219

Kummer variety, 178

lattice, 75Lefschetz trace formula, 271, 272left translation, 6line bundle

anti-symmetric, 21definition of, 1geometric, 1symmetric, 21, 178, 209totally symmetric, 178

linear equivalence of divisors, 8

morphismpurely inseparable, 73

multiplication by n, 74

Newton polygonof a crystal or isocrystal, 257

Newton slopes, 257nondegenerate

hermitian form, 312normalized symmetry, 178

odd theta characteristic, 228opposite ring, 308ordinary

elliptic curve, 81–82ordinary abelian variety, 262

p-rank, 80–81period

of a central simple algebra, 309Poincare bundle, 224polynomial ring

skew, 245positive involution, 313positivity

of the Rosati involution, 271prime

of a number field, 2principal divisor, 8purely inseparable morphism, 73

q-Weil number, 289conjugate, 289

quasi-isogeny, 181

reduced characteristic polynomial, 311

reduced norm, 311

reduced trace, 311

relative Cartier divisor, 222–223

Riemann hypothesisfor varieties over a finite field, 270

Riemann-Rochfor curves, 9

right translation, 6

Rigidity lemma, 12

semilinear map, 245

semisimplemodule, 308ring, 308

!a-F -crystal, see crystal

!a-F -isocrystal, see isocrystal

simplemodule, 308ring, 308

skew polynomial ring, 245

slope decomposition, 256

splitting field, 309

supersingularelliptic curve, 81–83

supersingular abelian variety, 262

symmetric line bundle, 21, 178, 209

symmetric power, 77, 221

symmetric tensors, 76, 79

symmetrizer map, 76

theta characteristic, 228even, 228odd, 228

theta divisordefinition of, 225–230

torsion points, 74–75, 80–86density of, 83–86

torsor, 174trivial, 175

totally symmetric line bundle, 178

translation, 6

variety, 1

vector bundledefinition of, 1geometric, 1

Verschiebung, 76–80definition, 78isogeny, 80iterated, 80

– 322 –

Weierstrass equation, 81

Weil conjectures, 270

for a curve, 273

for an abelian variety, 266–271

Weil number, see q-Weil number

Zeta function, see also Weil conjecturesdefinition of, 268of an abelian variety over a finite field, 269

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