Silverman J.H. — The arithmetic of elliptic curves :: Электронная библиотека попечительского совета мехмата МГУ

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Silverman J.H. — The arithmetic of elliptic curves

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Название: The arithmetic of elliptic curves

Автор: Silverman J.H.

Язык:

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

Siegel, C.L.      244 255 266 269 270
Sigma function      see Weierstrass $\sigma$ -function
Sign of the functional equation      362
Silverman, J.H.      368
Simply transitive group action      287
Singular point      18 20 48 61
Singular point of a scheme      357 358
SIZE      258
Smooth      see Non-singular
Spec(R)      184 357—359
Special fiber      184 357—360 361
Special value      362
Specialization homomorphism      367—368
Split multiplicative reduction      see Reduction of an elliptic curve
Split quaternion algebra      102 108
Standard absolute values      190 206
Stark, H.      268 340
Stokes' theorem      150
Subtraction map      288 289
Summation map      68 84 153 295
Sup norm      257 258
Supersingular      137 140 144 145 309;
Support of a divisor      43 108
Tamely ramified      188
Tangent line      14 19 43 49 55 58 61 62 104 106 240 357
Tangent plane      18
Taniyama — Weil conjecture      355 362
Tate curve      338 355—357
Tate module       90—95 134 145
Tate module as a $\mathbb{Z}_l$ -module      91
Tate module as a homology group      94
Tate module , in characteristic/)      91 139 145
Tate module , action of $G_{\bar{K}/K}$       91 94 95 109 178 179 188 273 366
Tate module , action of inertia      184 186 187 361
Tate module , of $K(T_l(\mu))$       91 135
Tate module , Weil pairing on      99 135
Tate — Shafarevich group      see Shafarevich — Tate group
Tate's algorithm      181 227 338 357 360
Tate, J.      172 184 228 229 234 357 363 364 371
Taylor series      48 234 270
Theorem 90      see Hilbert theorem 90
Theta function      see Weierstrass $\sigma$ -function
Thue equation      273
Thue, A.      244 270 273 371
Torsion conjecture      223
Torsion point      95; see also m-torsion subgroup
Torsion point over global fields      176 178 187 220—223
Torsion point over local fields      175—178
Torsion point, $\Leftrightarrow \hat{h}=0$       229
Torsion point, associated moduli problem      352—355
Torsion point, generates abelian extensions      341 342
Torsion point, integrality conditions      177 220—221 237
Torsion point, one-parameter family      223 238;
Torsion subgroup $(E_{tors})$       73 189 220 233 277 362;
Torsion subgroup $(E_{tors})$ of a CM curve      341 342
Torsion subgroup $(E_{tors})$ over $\mathbb{Q}$       223 238 239 262 323
Torsion subgroup $(E_{tors})$ , computation of      176 222 275 282 311
Torsion subgroup $(E_{tors})$ , p-primary part      223
Torus      148 149 159 160 342 346
Trace      41 101 134
Trace of Frobenius      135
Transcendence degree      8 26 79 165 286
Transcendence theory      241 274
Transcendental number      243 256 257 272
Translation map      68 75 76 80 97 109 284 287 292 293 300 306
Triangle inequality      209 211 271
Triplication formula      72 104
Trivial homogeneous space      290 297
Twist by a quadratic character      287
Twist of a curve      284—287
Twist of a function field by a cocycle      286 287 293
Twist of an elliptic curve      239 284 287 293 306—309 319 322 323 329 357
Twist of an L-series      362
Twisted product of groups      306 329 337
Two-descent      281
Two-isogeny      see Isogeny of degree two
Two-sphere      148
Uniformization theorem      150 161—164 349 355 356
Uniformization theorem, non-archimedean      356 357
Uniformizer      22 23 28 30 35—37 44 53 110 171 246
Unique factorization domain (UFD)      16
Unit equation      see S-unit equation
Unit group      33 171 190 232 234 235 263 269;
Unramified at P      28
Unramified at v      178 179 184 198 236 280 298 322
Unramified extension of fields      124 178 181 185 188 193 280
Unramified map      28 76 79 107 251 252

Unramified outside S      194 196 280 299
Upper half-plane $(\mathbb{H})$       343—345 349
Upper half-plane $(\mathbb{H})$ , extended $(\mathbb{H}^*)$       349 350 351 353 354
v-adic analytic map      356
v-adic distance function      see Distance function
v-adic topology      187 245 247 252 356 364 365
Valuation of a local coordinate ring      22 32;
Valuation of the minimal discriminant      172 183
Valuation, associated to an absolute value      189 280
Variety      1 5—20 367
Variety over a finite field      132—134
Vojta, P.      268
Weak Mordell — Weil theorem      116 175 189 190—196 220 235 251 276 278 287 298;
Weber function      341 342
Weber, H.      339
Weierstrass $\sigma$ -function      156 166 346 365
Weierstrass $\wp$ -function      153—162 262
Weierstrass $\wp$ -function as a modular function      346
Weierstrass $\wp$ -function generates elliptic function field      154
Weierstrass $\wp$ -function, algebraic relation      157 158
Weierstrass $\wp$ -function, Fourier series      346
Weierstrass $\wp$ -function, Laurent series      157
Weierstrass $\wp$ -function, relation with a      156 166
Weierstrass $\zeta$ -function      166
Weierstrass class of an elliptic curve $(\bar{a}_{E/K})$       224 225 238
Weierstrass coordinate functions      63 64 65 69 81 82 87 106 110 161 218 220 248 249 256 279 329 341 365
Weierstrass equation      45 46—55 57 58 74 103 106 111 121 131 140 144 145 160 177 183 184 187 202 216 262 263 268 275 280
Weierstrass equation is an elliptic curve      64
Weierstrass equation of a twist      287 308
Weierstrass equation, affine      46 241
Weierstrass equation, composition law on      55
Weierstrass equation, Deuringform      109 327
Weierstrass equation, discriminant      48 50 60 61 72 104 180 183 223 227 239 263 294 324 325 327 365
Weierstrass equation, formal solution      114
Weierstrass equation, global minimal      224—227 238 240 362
Weierstrass equation, global minimal, calculation of      227
Weierstrass equation, global minimal, may not exist      226 238
Weierstrass equation, group law      55—63 65 66
Weierstrass equation, invariant differential      48 52 65 79—84 85 113 149 172 329
Weierstrass equation, j-invariant      see j-invariant
Weierstrass equation, Legendre form      53—55 108 141 143 147 167 168 182 183 327
Weierstrass equation, linear change of variables      49 53 61 64 172 173 224 225 324 364
Weierstrass equation, minimal      171—174 179 180 186 187 193 224 357 359 365
Weierstrass equation, non-singular      50 63 64 250 325
Weierstrass equation, non-singular part $(E_{ns})$       60—63 104 173
Weierstrass equation, normal forms      46 48 324
Weierstrass equation, point at infinity      46 50 55 132
Weierstrass equation, quasi-minimal      238 251
Weierstrass equation, real locus      48 167
Weierstrass equation, reduction modulo $\pi$       171 173;
Weierstrass equation, S-minimal      227 264
Weierstrass equation, singular      48 53 60—63 104
Weierstrass preparation theorem      129
Weierstrass, K.T.W.      153
Weight of a modular function or form      153 344 345 347 348
Weil Conjectures      132—134 348
Weil conjectures for $\mathbb{P}^n$       144
Weil conjectures for elliptic curves      134—136
Weil curve      355 361 362 363
Weil pairing      95—100 107 108 277 278 279 320 353
Weil pairing, alternative definition      107 319
Weil pairing, dual isogeny is adjoint      98
Weil pairing, image of      97
Weil pairing, l-adic      99 135
Weil pairing, relative to an isogeny      107 319
Weil reciprocity law      43 108 370
Weil — Chatelet group      290 291 294 296 297 300 302 307 320;
Weil — Chatelet group over $\mathbb{R}$       320
Weil — Chatelet group over a finite field      320
Weil — Chatelet group, group law      291 319
Weil — Chatelet group, Tate pairing      364
Weil — Taniyama conjecture      see Taniyama — Weil conjecture
Weil, A.      81 132 134 361 362
Wiistholz, G.      263
Wild ramification      361
Wiles, A.      363
Winding number      152
Wronskian determinant      270
Zagier, D.      363
Zero isogeny      57 70 71
Zero of a function      22 38 151 152 155 157 245
Zeta function of $\mathbb{P}^n$       133
Zeta function of a variety      133—134
Zeta function of an elliptic curve      136 360
Zeta function of Riemann      166 236 345
Zeta function of Weierstrass      166
Zimmer, H.      371
Zorn's lemma      165

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