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Silverman J.H. — The arithmetic of elliptic curves
Silverman J.H. — The arithmetic of elliptic curves

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Название: The arithmetic of elliptic curves

Автор: Silverman J.H.

Язык: en

Рубрика: Математика/Алгебра/Алгебраическая геометрия/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1986

Количество страниц: 402

Добавлена в каталог: 12.03.2005

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Предметный указатель
$0^{th}$ cohomology group      see $H^0$
$1^{st}$ cohomology group      see $H^1$
$E/E_0$      183—185 187 358 359
$e_m$-pairing      see Weil pairing
$E_n$      187
$G_{\bar{K}/K}$ acting on $\mathscr{L}(D)$      40
$G_{\bar{K}/K}$ acting on affine coordinate ring      7 20
$G_{\bar{K}/K}$ acting on affine space      6
$G_{\bar{K}/K}$ acting on an algebraic group      295 333 336
$G_{\bar{K}/K}$ acting on Aut(E)      103
$G_{\bar{K}/K}$ acting on divisor group      31
$G_{\bar{K}/K}$ acting on function field      7 32
$G_{\bar{K}/K}$ acting on maps      15 20 284—286
$G_{\bar{K}/K}$ acting on Picard group      32 295
$G_{\bar{K}/K}$ acting on projective space      10 20
$G_{\bar{K}/K}$ acting on Tate curve      356 357
$G_{\bar{K}/K}$ acting on Tate module      91 94 95 109 178 179 188 273 341 366
$G_{\bar{K}/K}$ acting on torsion points      90 178 179 265
$G_{\bar{K}/K}$ acting on twisted function field      286 287 309
$G_{\bar{K}/K}$ acting on varieties      6
$G_{\bar{K}/K}$ acting on vector space      40
$G_{\bar{K}/K}$-module      197 198 295 298 333—337 356 357
$G_{\bar{K}/K}$-module, invariant elements      334
$G_{\bar{K}/K}$-module, unramified at v      198
$H^0$      3 330—337
$H^1$      3 197 198 330—337
$H^1$ = Horn if trivial action      331 334
$H^1$ of $E[\varphi]$      296—300; see also Selmer group
$H^1$ of $GL_n$      336
$H^1$ of $\hat{K}*$      20 43 198 277 279 321 335 336
$H^1$ of $\hat{K}^+$      335
$H^1$ of $\mu_m$      198 199 277 300 308 320 335
$H^1$ of Aut(E)      307—309 319 322 329
$H^1$ of E      197 287 291 294 296 297 307;
$H^1$ of E[m]      197 287 320;
$H^1$ of Isom(E)      285 292 306 307
$H^1$, unramified outside S      299
$M_2$      94 102 108
$M_K$      see Absolute values
$SL_2(\mathbb{Z})$      343 344 346 349—354 362
$\mathscr{L}(D)$, l(D)      see Divisor associated
$\tau$-function of Ramanujan      345 348
$\varphi$-approximable      272
$\varphi$-Selmer group      see Selmer group
$\wp$-function      see Weierstrass $\wp$-function
$\wp^1$      32 37 39 42 53 65 358
Abelian extension      95 188 193 194 196 236 299 338 339 341 342
Abelian group      4 55 88 131 174 197 199 323 330 333
Abelian variety      2 94 295 342 358 363
Absolute values $(M_K)$      189 243 277
Absolute values $(M_K)$, archimedean $(M_K^{\infty})$      189 250
Absolute values $(M_K)$, non-archimedean $(M_K^0)$      189
Absolute values $(M_K)$, standard      190 206
Absolute values $(M_K)$, valuation associated to $(ord_v)$      189 195
Addition formula      58 80 81 82 106 110 187 202 216
Addition formula, formal      114
Addition law      see Group law
Additive reduction      see Reduction of an elliptic curve
Affine algebraic set      6
Affine coordinate ring      7 20
Affine n-space      5 8
Affine variety      7 13 26 44
Affine Weierstrass equation      46 241
Algebraic group      333 336 358
Algebraic set      see Affine algebraic set; Projective algebraic set
Analytic continuation      361 362
Analytic map      see Complex analytic map; v-adic analytic map
Approximation exponent      243 244
Arclength of a lemniscate      169
Arclength of an ellipse      146 149 169
Arithmetic intersection theory      366
Arithmetic-geometric mean      169—170
Artin map      340 342
Artin — Schreier extension      329
Artin, E.      130
Artin, M.      134
Associative law for a formal group      115 120
Associative law for a homogeneous space      288
Associative law for an elliptic curve      55 57
Automorphism group (Aut)      71 103 107 284 306—309 325 329 341
Automorphism group (Aut) as a $G_{\bar{K}/K}$-module      103 308
Automorphism group (Aut), action on WC group      319
Automorphism group (Aut)of Tate module      95 366
Auxiliary polynomial      270
Bad reduction      see Reduction of an elliptic curve
Baker's theorem      257 260
Baker's theorem, p-adic version      261
Baker, A.      257 261 340
Basepoint of an elliptic curve      42 45 46 306
Bernoulli polynomial      366
Bezout's theorem      55
Big-O notation      166 215
Bilinear pairing      88 131 191 229 232 278 306 319 323 364
Birational      53
Birch and Swinnerton — Dyer conjecture      234 315 362 363
Branch cut      148 149 168 346
Brauer group      102 108
Brauer — Hasse — Noether theorem      102 321
Brauer — Siegel theorem      269
Bremner, A.      235
Canonical divisor      37—40
Canonical height      see Height; see also Neron — Tate pairing
Cassels' pairing      306 315 363
Cassels, J.W.S.      177 231 235 306 363 364
Cauchy sequence      228
Cauchy — Schwarz inequality      131
Central simple algebra      102
Chain rule      120
CHARACTER      132
Character, sum over a finite field      132 141
Characteristic p      89 126—128 130—145
Characteristic polynomial      135
Characteristic two and three      49 52 53 72 103 324—329
Characteristic zero      83 89 102 105 121 122 125 146 164 165 177 194
Chatelet, F.      371
Chinese remainder theorem      225
Class field theory      188 338 370
Class number      167 194 238 269
Class number, one      226 227 238 264 265 321 340
Coates, J.      261 363
Cohomologous one-cocycles      285 331 334 336
Cohomology class      198 199 285 331
Cohomology class, unramified      198 236
Cohomology group      291 330—337
Cohomology set      285 306 336
Complementary modulus      168
Complete elliptic integral      168—170
Complete ring      112 117—119 123 125
Complex analytic isomorphism      105 150 158 161 162 262 340 342 346 349 353 354
Complex analytic map      159 161 163
Complex Lie group      see Lie group
Complex multiplication      73 95 109 144 168 263 265 273 338—342 361 362 363 366 370 371
Complex multiplication $\Rightarrow$ potential good reduction      181 188
Complex multiplication for curves over Q      340
Composition law      55 60
Conductor of an elliptic curve      358 359 361 362
Conductor of an order      108
Congruence subgroup      350—351 353 354 362
Connecting homomorphism      197 198 199 277 278 300 319 332 334
Constant map      25
Continuous map      187 334 336
Continuous one-cocycle      334 336
Convex set      231
Coordinate functions      see Weierstrass coordinate functions
Coplanar points      106
Cotangent space      159
Curve      21—44
Curve, distance function on      245—247
Curve, hyperelliptic      26 44 255 293
Curve, maps between      23—30
Curve, modular      see Modular curve
Curve, smooth      see Non-singular curve
Curve, twists of      284—287
Cusp      49 50 60 61 104 180 240 349 350
Cusp form      344 345 347 348
Cusp form for a congruence subgroup      350 351
Cusp form of weight two      350 351 362
Cusp of a congruence subgroup      350 354
Cyclotomic representation      92
d-uple embedding      237
Davenport, H.      268
Decomposition group $(G_v)$      296
Dedekind $\eta$-function      346 350
Dedekind sum      346
Defined at P      9 15 16 22 35
Defined over K      6 11 14 15 21 31 40 46 57 60 63 71 94 208 287 289 300
Definite quaternion algebra      100
Degree      see also Inseparable degree; Local degree; Separable degree
Degree map, is positive definite      88 131
Degree of a curve      106
Degree of a divisor      31 32 152
Degree of a homogeneous polynomial      10
Degree of a map      25 42 246
Degree of a morphism      208
Degree of an algebraic number      242 243
Degree of an endomorphism      134 265
Degree of an isogeny      70 76 88
Degree of multiplication-by-m map      86 89 105 106 107 163
Degree of the zero isogeny      70
Degree, one      25 53 64 105 290
Dehomogenization      13 27 62
Deligne, P.      134 348
Dem’janenko, V. A.      371
Descent      189 199
Descent theorem      199 205 206
Descent via two-isogeny      302 303 311
Descent via [2]-map      281 282
Determinant      134
Deuring normal form      109 327
Deuring, M.      102 144 145 361
Differential form      34—37
Differential form, associated divisor      36
Differential form, associated to a cusp form      350 362
Differential form, holomorphic      36 37 39 44 52 159 329 350
Differential form, invariant      48 52 65 79—84 85 113 149 172 362
Differential form, non-vanishing      36 52 159
Differential form, order at P      36
Differential form, regular      see Differential form holomorphic
Differential operator      143 270
DIMENSION      8 14 18 21 133 295
Dimension of a hypersurface      20
Dimension Theorem      19
Diophantine approximation      241 242—245 247 251—254 265 274
Diophantine equation      1 12 241 244 249 257 291
Diophantine geometry      1 6 17 45
Diophantine inequality      245
Dirichlet series      234 240;
Dirichlet's S-unit theorem      195 253 255 258
Dirichlet, G.L.      242
Discrete $G_{\bar{K}/K}$-module      see $G_{\bar{K}/K}$-module
Discrete topology      333 334 336 364
Discrete valuation      192
Discrete valuation, ring      21 22 42 123—126 171
Discriminant as a modular form $(\Delta(\tau))$      343 345 347 348 356
Discriminant of a lattice      158 167 342 343 365
Discriminant of a number field      232 235 269
Discriminant of a polynomial      26 44 51 158 273
Discriminant of a Weierstrass equation      see Weierstrass equation discriminant
Discriminant, minimal $(\Delta_v, \mathscr{D}_{E/K})$      172 183 186 224 227 233—235 239 264 358 359 361
Disjoint support      see Support of a divisor
Distance function      245—247 248 250 251
Distance function, effect of a finite map      246
Distributive law      71
Divisible element      306 364
Division polynomial      105 177
Divisor      31—34
Divisor class group      32; see also Picard group
Divisor function      345
Divisor group (Div)      31 37 44 66 68 84 152 295 321
Divisor inequality      38
Divisor of a differential      36
Divisor support      43 108
Divisor, associated to a differential      36
Divisor, associated to a function      32 153 280 295
Divisor, associated vector space $(\mathscr{L}(D))$      38 40 42 64 66 104 106
Divisor, canonical      37—40
Divisor, defined over K      31 40 44 68
Divisor, degree      31 38 152
Divisor, degree 0      31 32 152
Divisor, evaluated at a function      43 108
Divisor, linear equivalence      32 38 65
Divisor, positive (effective)      37 322
Divisor, principal      32 67
Dual isogeny      73 74 84—90 168 301 302 310
Dual isogeny is adjoint for Weil pairing      98
Dual isogeny of Frobenius      137
Dual isogeny of multiplication-by-m      86
Dual isogeny, definition      86
Duality      364
Duplication formula      59 72 104 203 217 221 222 310
Dwork, B.      134
Dyson, F.      244 270
Effective divisor      see Divisor
Effectivity      196 201 241 245 248 250 252 254—263 271 276 279 304
Eichler, M.      145 361
Eigenfunctions      348 351
Eisenstein series      153 342—345 347
Elimination theory      211
ellipse      1 146 149 169
Elliptic curve      2 14 21 45—368
Elliptic curve in characteristic two and three      49 52 53 72 324—329
Elliptic curve, $\mathbb{R}$      48 167 275 362
Elliptic curve, defined over K      46 63 71 307
Elliptic curve, definition      63
Elliptic curve, finite fields      60 130—145 323
Elliptic curve, group of rational points is a subgroup      57
Elliptic curve, group of rational points over $\mathbb{C}$      146—170
Elliptic curve, integral points      59 241—275
Elliptic curve, K-isomorphism classes      308
Elliptic curve, local fields      171—188 355—357
Elliptic curve, number fields      189—240 276—323
Elliptic exponential      262
Elliptic function      3 146 150—159 161 346 355;
Elliptic function as function of $\wp$ and $\wp'$      154
Elliptic function, as product of $\sigma$-functions      156
Elliptic function, field      76 150 154
Elliptic function, no poles $\Rightarrow$ constant      151 160
Elliptic integral      146 147—150 168—170 370
Elliptic logarithm      262—263
Elliptic regulator      233 234 362
Elliptic regulator is positive      233
Elliptic surface      143 368
Endomorphism ring (End)      71 100—102 103 109 134 163 168 339—341
Endomorphism ring (End) is an integral domain      72 88 100
Endomorphism ring (End), $e_{\varphi}$-pairing      107; see also Weil pairing
Endomorphism ring (End), classification of      102 137 145 164 165
Equivalence of categories      26 162
Equivalence of homogeneous spaces      290 291
Euclidean algorithm      1 204
Euler characteristic      134 136 144
Euler product      240 361
Even function      59 154 155 216 218—220 227—229 247 248
Evertse, J.-H.      254
Exponent m, Galois group with      193 194 196 236 299
Exponent of the conductor of an elliptic curve      358 359 361
Exponential of a formal group      121
Extended upper half-plane $(\mathbb{H}^*)$      349—351 353 354;
Extension formula      206
Faltings, G.      94 266 366
Families of elliptic curves      223 234 238 276 309 323
1 2 3 4
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