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

Fermat's last theorem      7
Fermat, P.      266
Fiber      184 357—361
Finite map      25 75
Finitely generated group      189 199 200 201 220 241 254
Fixed field      286
Formal addition law      114
Formal derivative      120
Formal group      110 115—129 177
Formal group in characteristic 0 is commutative      122
Formal group in characteristic p      126—128
Formal group over a DVR      123—126
Formal group, additive group $(\hat{\mathbb{G}}_a)$       116 118 119 124 126
Formal group, associated group      117—119 123—126
Formal group, associated to an elliptic curve (E)      115 116 118 121 128 137 174 176 177 183 184
Formal group, associative law      115 120
Formal group, defined over R      115
Formal group, exponential $(exp_{\mathscr{F}})$       121—123 126
Formal group, height      126 129 137
Formal group, homomorphism      115 120 126
Formal group, invariant differential      119—121 122 127
Formal group, isomorphic      116 122
Formal group, law      115 117 120 129
Formal group, logarithm $(log_{\mathscr{F}})$       121—123 124 126 184
Formal group, multiplication-by-m map      116
Formal group, multiplication-by-p) map      120
Formal group, multiplicative group $(\hat{\mathbb{G}}_m)$       116 118 119 121
Formal group, non-commutative      129
Formal group, torsion of, over $\mathbb{Z}_p$       124
Formal group, torsion points      118 123 124 126 129 177
Fourier coefficients      345
Fourier series      344 345 355 356
Fractional ideal      33 237 341
Free product of groups      343
Frobenius (endo)morphism      19 29—31 74 83 85 89 128 131 135 137 141 145 320
Frobenius (endo)morphism is purely inseparable      30
Frobenius (endo)morphism, 1-, is separable      83 131
Frobenius (endo)morphism, characteristic polynomial      135
Frobenius (endo)morphism, degree of      30
Frobenius (endo)morphism, dual of      137
Frobenius element      341
Fueter, R.      339 371
Function field      7 14 15 21 26
Function field as base field      367—368
Function field, elliptic      see Elliptic function field
Function field, map induced on, by rational map      24 286
Function field, twisted by a cocycle      286 287 293 309
Function, analytic      342
Function, defined by a rational map      24 33 215
Function, defined P      9 15 22 35
Function, elliptic      see Elliptic function
Function, even      59 154 155 216 218—220 227—229 247 248
Function, modular      see Modular function
Function, no poles $\Rightarrow$ constant      22 151 160
Function, odd      155 219
Function, value at a divisor      43 108
Functional equation      134 136 137 144 361 362
Fundamental domain      232 233 243
Fundamental parallelogram      150—152 262
Fundamental parallelogram, area of      166
Fundamental parallelogram, boundary of      151
Fundamental parallelogram, closure of      150
G-invariant element      330 334 336
G-module      330—333
G-module, exact sequence of      331
G-module, homomorphism      330
GAGA      163
Galois cohomology      197 198 296 321 333—337
Gauss, C.F.      170 316 318
Gelfond, A.O.      244 256 257 270
General linear group       44
Generic fiber      184 357 358
Genus      21 39 41 79 104 252 266 284 295
Genus of $\mathbb{P}^1$       39
Genus of a hyperelliptic curve      44
Genus of a modular curve      351 355
Genus of a quotient curve      79 107
Genus of a smooth plane curve      43
Genus, formula      see Hurwitz genus formula
Genus, one      2 21 40 42 44—46 63—66 104 108 109 248 256 261 294 319—321 351 360
Genus, zero      2 39 42 64
Global minimal Weierstrass equation      see Weierstrass equation
Good reduction      see Reduction of an elliptic curve
Graded ring      111 347
Gram — Schmidt      274
Green's function      370
Greenberg, R.      263
Gross, B.      363
Grothendieck, A.      134
Group associated to a formal group      see Formal group
Group cohomology      3 191 196 277 330—337;
Group cohomology, non-abelian      335—336
Group law on an elliptic curve      55 57 58 60 65 80 81 354
Group law on an elliptic curve is a morphism      68
Group law on an elliptic curve, effect on height      216
Group scheme      184 358
Hall, M.      268
Hasse invariant      137 140 145;
Hasse principle      2 12 234 276 277 279 360
Hasse — Minkowski theorem      2
Hasse, H.      75 131 132 341
Hecke L-series      361
Hecke operator      347 348 351 362
Heegner point      363
Heegner, K.      340
Height      189 190 201 205—220 227—233 241
Height of a formal group      126 128 129 137
Height of a homomorphism of formal groups      126 128
Height of a rational number      202
Height of an algebraic number      211
Height of an algebraic number, equals 1 $\Leftrightarrow$ root of unity      236
Height of an algebraic number, relation to size      258
Height of generators for Mordell — Weil group      235
Height of integral points      250 261 263 268
Height of the roots of a polynomial      211
Height on $\mathbb{P}^n(\mathbb{Q})$       206 207
Height on an abelian group      199
Height on an elliptic curve      201 215—220 227—233 239 251 305
Height on projective space      205 207 215 236
Height, absolute      208 215 252
Height, action of $G_{\bar{K}/K}$ on      213
Height, behavior under maps      208 215 216 219 229 237 239 275
Height, canonical $(\hat{h})$       219 227—233 235 262 363—365;
Height, canonical $(\hat{h})$ , computation of      364
Height, canonical $(\hat{h})$ , difference from usual height      229 239
Height, canonical $(\hat{h})$ , equals 0 $\Leftrightarrow$ torsion point      229
Height, canonical $(\hat{h})$ , is positive definite      232 262
Height, canonical $(\hat{h})$ , lower bound      233 239 274
Height, finitely many points with bounded      200 202 206 213 216 236
Height, local      228 364—366
Height, logarithmic      215 235 257 269
Height, Neron — Tate      see Height canonical
Height, relative to       215 220 227—230 247 250—252 275
Height, relative to       207 241 243 245 247 252 258 259 271
Hensel's lemma      2 111 112 174 297 299 312 322
Hessian matrix      106
Hilbert basis theorem      6 165
Hilbert class field      167 339 340 341
Hilbert irreducibility theorem      367
Hilbert problem      256
Hilbert theorem ninety      20 43 198 277 279 321 335 336
Holomorphic differential      36 37 39 44 52 159
Holomorphic differential on $\mathbb{P}^1$       37
Homogeneous ideal      10 11 13 29
Homogeneous polynomial      10 15 16 20 30 133 203 204 208 273
Homogeneous space      276 277 279 284 287—296 297 299 304 310 319;
Homogeneous space is a twist      289
Homogeneous space, associated to a quadratic extension      293 301 302
Homogeneous space, divisor group      295 321
Homogeneous space, equivalence of      290 291

Homogeneous space, index      321—322
Homogeneous space, locally trivial      298 304 311 312 316 322
Homogeneous space, period      321—322
Homogeneous space, Picard group      294 295 321
Homogeneous space, summation map on      295
Homogeneous space, trivial      290 297
Homogeneous, coordinates      10 20 43 61 133 173 205 207 213
Homogenization      13
Homology of $E(H_1(E,\mathbb{Z}))$       149 161
Homomorphism group (Horn)      71 81 92 107 145;
Homomorphism group (Horn) is a torsion-free $\mathbb{Z}$ -module      71 93
Homomorphism group (Horn) of Tate modules      92 94 107 145 366
Homomorphism group (Horn), degree map on      88 92
Homomorphism group (Horn), rank $\leq$ 4      94
Homomorphism of formal groups      115 120 126
Homothetic lattices      161 163 164 342 343
Homothety      see Scalar multiplication
Hurwitz genus formula      41 42 44 65 74
Hurwitz, A.      242 370
Hyperbola      1
Hyperelliptic curve      26 44 255 293
Hyperplane      11 12 25 106 258
Hypersurface      20
Ideal class group      33 194 224 235 339 340
Ideal of a variety      6
Identity component      358
Index of a homogeneous space      321—322
Infinite descent      189
Inflation map on cohomology      332 335 337
Inflation-restriction sequence      197 236 299 332 335 337
Inseparable degree      25 70 76 128
Inseparable map      25 137
Integral extension of rings      249
Integral j-invariant      see j-invariant
Integral points      3 59 241—275
Integral points in $\mathbb{Z}$       3 59 260 261 266 268 272 273 275
Integral points number of      250 251 272
Integral points on hyperelliptic curves      255
Integral points, effective bounds      252 261 263 268
Integral points, finitely many S-      248 249 252 255 266 273
Invariant differential      48 52 65 79—84 85 113 149 172
Invariant differential of a formal group      119—121
Invariant of a quaternion algebra      102
Inverse limit      333
Isogenous elliptic curves      70 145 168 185 363
Isogenous elliptic curves have same bad primes      185 299
Isogenous elliptic curves have same coductor      361
Isogenous elliptic curves, finitely many over K      264 265
Isogeny      70—79 81 84 92 276;
Isogeny is a homomorphism      75 161
Isogeny of degree two      74 301 302 310 311
Isogeny, associated moduli problem      352—354
Isogeny, defined over K      71 94 264 265 296 298
Isogeny, Frobenius      see Frobenius (endo)-morphism
Isogeny, given by an analytic map      159—161 162
Isogeny, inseparable degree      128
Isogeny, kernel of      76 107 265 296 301 302 311 319 320 352
Isogeny, separable      76 78 85
Isogeny, theorem      94
Isogeny, Weil pairing on kernel      107 319
Isomorphism      17 19
Isomorphism group (Isom)      107 284—286 292 306 307;
Isomorphism group (Isom) may be non-abelian      285
Isomorphism group (Isom) of an elliptic curve      306 307
Isomorphism of curves      25 264 284 300
Isomorphism of formal groups      116 122
Isomorphism, complex analytic      105 150 158 161 162
Isomorphism, defined over K      17
J-invariant      48—52 54 137 140 143 168 183 233 239 308 324 325 327 339 349 359 366
j-invariant as a modular function      343 345 349 356
j-invariant of a CM elliptic curve      188 339 341 342
j-invariant of a curve of genus one      108 109
j-invariant of a lattice      167 339 342 343
j-invariant, integral      181 186 188 233 251 328 339
j-invariant, integral $\Rightarrow$ potential good reduction      181 328
j-invariant, j=0 and j=1728      54 103 104 144 145 167 308 309 323 325 329 341
j-invariant, non-integral      356 357
j-invariant, transcendental      137 145
Jacobi, C.G.J.      345
Jacobian variety      294 367
Jordan normal form      136
K-rational points      5 6 10 11 20
Kani, E.      370
Katz, N.      177
Kenku, M.      265
Kodaira symbol      359
Kodaira, K.      183 358
Kronecker — Weber Theorem      338
Kronecker's theorem      236
Krull's Hauptidealsatz      20
Krull's theorem      119
Kummer pairing      191 196 197 277 279
Kummer pairing via cohomology      196—199
Kummer pairing, kernel of      191 277
Kummer sequence for E      197 278 287
Kummer sequence for is: $\bar{K}*$       198 278
Kummer theory      194 196
l-adic cohomology      134
l-adic representation      91 187
l-adic representation is irreducible      273
l-adic representation, image of      95 366
l-adic Tate module      see Tate module
l-adic Weil pairing      see Weil pairing l-adic
L-series      234 240 338 360—363;
Land, S.      177 233—235 250 254 268 371
Lang — Trotter conjecture      144
Laska, M.      227
Lattice      105 149 150 153 159 160 162 163 165 166 3\231 261 265 274 342 349
Lattice discriminant      158 167 342 343 345
Lattice homothetic      161 163 164 342 343
Lattice in $\mathbb{R}\bigotimes E(K)$       231 232
Lattice of an ideal      339
Lattice, j-invariant      167 339 342
Lattice, rectangular      167
Laurent series      110 113 157
Lefschetz principle      164 165
Legendre normal form      53—55 108 141 143 167 168 182 183 327
Legendre relation      166
Legendre symbol      317
Lehmer, D.H.      144
Lemniscate      169
Lie group      158 161 162
Limit formula      363
Lind, C.-E.      304 316
Line      1 11 14 19 27 55 67 114 174
Line at infinity      46 49
Line integral      146 147;
Linear change of variable      1 49 53 61 64 106 172 173 224 324
Linear equivalence      32 38 65
Linear forms in elliptic logarithms      262—263 269
Linear forms in logarithms      245 248 252 257 259 260
Linear forms inp-adic logarithms      261
Linear fractional transformation      343
Liouville's theorem      151 272
Liouville, J.      242 244
Local class field theory      188
Local degree at       206 241 258
Local height function      see Height function
Local ring at P      9 15 21
Local ring at P is a DVR      21 42
Locally trivial homogeneous space      298 304 311 312 316 322;
Logarithm of a formal group      121—123
Logarithm, elliptic      262—263
Logarithm, linear form      see Linear forms in logarithms
Logarithm, p-adic      257 261
Lutz, E.      221
m-torsion subgroup (E[m])      73 86 89 95 137 163 165 191 197 277 278 320;
m-torsion subgroup (E[m]) is unramified      179 184
m-torsion subgroup (E[m]), action of $G_{\bar{K}/K}$       90 178 179 187 265
m-torsion subgroup (E[m]), associated moduli problem      352—355

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