 169
 62
 171
 91
 38
 38
 1
 63
 ,  ,  ,  51
 41
 94
 41
 143
 177
 168
 8 60
 13 60
A* 166
A-field 18
Ab(X) 147
Abelian integral 178
Adele 62
Aff(X,F) 132
Affine 131
Affine K-variety 120
Affine K/k-variety 120
Affine piece 82 132
Algebraic function field 17
Algebraic manifold 146
Algebraically equivalent 139
Analytic ramification index 95
Archimidean 20
Artin — Whaples formula 44
B* 168
Base change 185
Base space 130
Birational map 81
Birationally equivalent 80 137
Blowing-up 86
Branch 89
C*(V) 139
Canonical sheaf 164
Castelnuovo's formula 175
Cauchy filter 35
Cauchy sequence 20
Cauchy's theorem 97
Cech nerve 148
Chow ring 140
Chow's theorem 146
Class number 8
Codimension 121
Coefficient group 192
Cohomology group 147 191
complete 20
Complex multiplication 114
Complex torus 107
Component 121 192
Conjugate module 151
Conjugate varieties 135
Conjugation 116
Connected 192
Continuous functor ' 138
Correspondence 174
Cot(V) 174
Cotangent space 130
Covering 190
Covering space 104
CYCLE 139
de Rham's theorem 166
Dedekind domain 7 54 78
Deg 48
Degree (of a spot) 48
Degree (of an ideal) 61
Derivation 160
Derivative, exterior 163 164
Differential 163 164 165
Dilation 86
DIMENSION 121
Dirichlet series 1 8
Dirichlet's unit theorem 6 55
Discrete spot 20
Discriminant 9 47
Div(x) 82
Division algebra 116
Divisor 58 102 139
Divisor class group 58 103
Divisor function 4
Dominating 137
Elliptic curve 107 116 195
Elliptic modular function 112
Espace etale 128
Etale 146
extends 26
Fibre 155
Field of constants 17
Fine 147
Finite type 182
Flabby 147
Fr(X) 56
Frac(B) 56
Fractional ideal 56
Fractional Laurent series 87
Frobenius map 79
Function field 17
Functional equation 3 9 14 15 61 143 153
Fundamental class 150
Fundamental group 105
Fundamental set of units 6
Genus 15 48 101
Geometric point 183
Global field 18 43 52
Grassmanian 154
Hansel's lemma 37
Harmonic form 168 171
Hermitian metric 170
Hilbert's basis theorem 85
Hilbert's Nullstellensatz 120 179
Hilbert's weak Nullstellensatz 124
Hodge's theorem 168 171
Homotopy type, Cech 193
Homotopy type, etale 193
Homotopy type, singular 196
Homotopy type, Verdict 193
HSS 148
Hurwitz's theorem 4
Hypercovering 193
ICG(B) 57
Ideal class group 7 13 56
Idele 63
Imaginary square root 12 29
Index theorem 101
Injective 147
Integers 32
Integral closure 6 17 34 79
Integral scheme 182
Intersection number 125 145
Irreducible 74 120
Irreducible components 121
Isogeny 117
J(X) 103 142 190
Jacobdan 103 142 177
Jacobian matrix 82 85 145
k-topology' 120 135
K/k-prevariety 131
K/k-variety 120 134
Kaehler manifold 172
Kaehler metric 172
l-adic cohomology 195
la(A) 131
| Laplace operator 168
Laplace — Beltrami operator 171
Lattice 107
Laurent series 1!
Lefschetz formula 151 195
Lies over 75 83
Lifting 183 187
Linearly equivalent 166
Liouville's theorem 96
Local homomorphism 32
Local parameter 94
Locally affine 131
Locally connected 192
Locally free 57 159
Luroth's theorem 17
m(X) 96
Meromorphic function 94
MOD 49
Model 18 85
Modular function 49
Modular group 112
Moebius function 4
Moebius transformation 108
Moving Lemma 140
N(I) 8
Nakayama's lemma 77
Noetherian scheme 182
Noetherian space 120
Non-archimidean 20
Norm 8
normal 126 134 182.
Normal scheme 182
Normalised valuations 43
Numerically equivalent 141 156
ord 21 53
P(F) 43
p-adic integers 41
p-adic numbers 41
p-adic spot 22
p-adic valuation 22
Parametrisation 89 99
Patron's formula 2
Periodic functions 108
Pic(B) 157
Place 89
Plane curve 74
Poincare duality 149 194
Poincare lemma 166 170
Point 183
Poisson summation formula 68
Pole 94
Pontrjagin duality 64
Pr(X) 103
Prescheme 181
PreSh(X) 158
Presheaf 127 191
Prevariety 131 186
Prime element 34
Principal divisor 58 142
Principal parts 163
Pro-HSS 148
Pro-object 148
Projective curve 82 85
Projective formula 141 149
Projective line 24 182
Projective space 81 182
Projective variety 137
Proper 197
Puisceux's theorem 88
Quadratic number field 10 11 47
Quasi-projective 137
R-Mod 158
R-Top 130
Ramification index 38 95 100
Ramified function 95
Rat(K/k) 137
Rational equivalence 140
Rational function 74 83 121 134
Rational map 137
Real square root 12 29
Reduced scheme 182
Reduction 32 143 154 164 187
Regular function 121 123 135
Regulator 9
Relative degree 38
Residue field 32 122 130
Restricted product 63
Riemann classification theorem 108
Riemann hypothesis 4 16 115 144 173
Riemann surface 94
Riemann — Enriques theorem 196
Riemann — Hurwitz Theorem 108
Riemann — Roch theorem 175
Riemannian manifold 167
Riemannian metric 167
Ring of integers 31
Ringed space 130
RS 94
Sch(V) 183
Scheme 181
Separated 133 184
Sh(V) 158
Sheaf 127 191
Sheaf of sections 127
SIMPLE 77 82 125 134
singular 77 82 94 125 134
Site 190
Smooth 77 82
SMR 36
Spec 179
Specialisation 82 124
Spectrum 123
Spot 21 43 44
Stoke's theorem 168
Strong approximation theorem 53
Strong topology 145
Structure sheaf 130
Symmetric product 103
System of multiplicative representatives 36
Tangent 125
Tangent sheaf 161
Tangent space 125 130 162
Tare module 178
topology 190
Topos 191
Transcendence degree 17
Trivial sheaf 127
Ultrametric 20
Umkehr homomorphism 149
Uniformity 88
Unique factorisation 1
Universal coefficient theorem 149
Universal domain 124
Unramified 38
Unramified function 95
V(G) 74 82
V(J) 119
Valence 95
Valuation 20
Valuation ring 31
Value group 20
Vector field 163
Weak approximation theorem 45
Weierstrass elliptic function 108
Weil Conjectures 117 143 197
Z(z,K) 61
Z.(z,K/F) 13 61
Zariski topology 120 135 179
Zeta function 3 8 60 79
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