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References
Books referred to by abbreviation
BLR
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MAV
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Index
F -crystal, see crystal
F -isocrystal, see isocrystal
fixed point scheme, 71
flex point, 82
Frobenius, 76–80
arithmetic, 310
characteristic polynomial of, 266
geometric, 265
isogeny, 76
iterated, 80
2-torsion
of an elliptic curve, 81
3-torsion
of an elliptic curve, 82
Abel’s Theorem, 223–224
abelian varieties up to isogeny
category of, 181
abelian variety
definition, 6
over C, 75–76
arithmetic Frobenius, 310
G-torsor, 174
Gauss map, 221
geometric Frobenius, 265
geometric line bundle, 1
geometric vector bundle, 1
group
definition, 5
group variety
definition, 6
Brauer equivalence, 309
Brauer group, 309
canonical involution, 190, 311
Cartier duality
of Frobenius and Verschiebung, 79
central simple algebra, 309
characteristic polynomial of Frobenius, 266
complex abelian variety, 75–76
complex torus, 75
conjugate
q-Weil numbers, 289
crystal, 246–262
definition of, 247
effective, 247
isogeny, 247
crystalline cohomology, 251
cyclic algebra, 310
Hard Lefschetz Theorem, 271
Hasse-Weil bound, 273
Hasse-Weil Theorem, 266
Hasse-Weil-Serre bound, 273
height
of a σ a -F -crystal, 248
hermitian form, 312
Hodge Index Theorem, 271
Hodge numbers
of a crystal, 250
homomorphism
definition, 13
of abelian varieties, 12–14
degree
of a central simple algebra, 309
of a divisor, 8
of a line bundle, 219
divisibility
of X(k), 74
divisor
degree, 8
effective, 8
linear equivalence, 8
on a curve, 8
principal, 8
divisor class group, 8
effective
σ a -F -crystal, 247
σ a -F -isocrystal, 248
effective divisor, 8
effective relative Cartier divisor, see relative Cartier
divisor
elliptic curve, 9
even theta characteristic, 228
exponential map, 75
index
of a central simple algebra, 309
involution, 311
canonical, 190, 311
of the first kind, 311
of the second kind, 311
positive, 313
irreducible
module, 308
isoclinic, 254
isocrystal, 246–262
definition of, 247
effective, 248
isogenous abelian varieties, 75, 80
– 321 –
isogeny, 72–80
definition, 72
degree, 72, 73
equivalence relation, 75
Frobenius, 76
multiplication by n, 74
of σ a -F -crystals, 247
purely inseparable, 73–74
quasi-, 181
separable, 73–74
Verschiebung, 80
iterated Frobenius, 80, 265
iterated Verschiebung, 80
Jacobi’s Inversion Theorem, 224
Jacobian
definition of, 91, 219
Kummer variety, 178
lattice, 75
Lefschetz trace formula, 271, 272
left translation, 6
line bundle
anti-symmetric, 21
definition of, 1
geometric, 1
symmetric, 21, 178, 209
totally symmetric, 178
linear equivalence of divisors, 8
morphism
purely inseparable, 73
multiplication by n, 74
Newton polygon
of a crystal or isocrystal, 257
Newton slopes, 257
nondegenerate
hermitian form, 312
normalized symmetry, 178
odd theta characteristic, 228
opposite ring, 308
ordinary
elliptic curve, 81–82
ordinary abelian variety, 262
p-rank, 80–81
period
of a central simple algebra, 309
Poincaré bundle, 224
polynomial ring
skew, 245
positive involution, 313
positivity
of the Rosati involution, 271
prime
of a number field, 2
principal divisor, 8
purely inseparable morphism, 73
q-Weil number, 289
conjugate, 289
quasi-isogeny, 181
reduced characteristic polynomial, 311
reduced norm, 311
reduced trace, 311
relative Cartier divisor, 222–223
Riemann hypothesis
for varieties over a finite field, 270
Riemann-Roch
for curves, 9
right translation, 6
Rigidity lemma, 12
semilinear map, 245
semisimple
module, 308
ring, 308
σ a -F -crystal, see crystal
σ a -F -isocrystal, see isocrystal
simple
module, 308
ring, 308
skew polynomial ring, 245
slope decomposition, 256
splitting field, 309
supersingular
elliptic 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, 228
even, 228
odd, 228
theta divisor
definition of, 225–230
torsion points, 74–75, 80–86
density of, 83–86
torsor, 174
trivial, 175
totally symmetric line bundle, 178
translation, 6
variety, 1
vector bundle
definition of, 1
geometric, 1
Verschiebung, 76–80
definition, 78
isogeny, 80
iterated, 80
– 322 –
Weierstrass equation, 81
Weil number, see q-Weil number
Weil conjectures, 270
for a curve, 273
for an abelian variety, 266–271
Zeta function, see also Weil conjectures
definition of, 268
of an abelian variety over a finite field, 269
– 323 –