Silverman J.H. — Advanced Topics in the Arithmetic of Elliptic Curves :: Электронная библиотека попечительского совета мехмата МГУ

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Silverman J.H. — Advanced Topics in the Arithmetic of Elliptic Curves

Silverman J.H. — Advanced Topics in the Arithmetic of Elliptic Curves

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Название: Advanced Topics in the Arithmetic of Elliptic Curves

Автор: Silverman J.H.

Аннотация:

This book continues the treatment of the arithmetic theory of elliptic curves begun in the first volume. The book begins with the theory of elliptic and modular functions for the full modular group r(1), including a discussion of Hekcke operators and the "L"-series associated to cusp forms. This is followed by a detailed study of elliptic curves with complex multiplication, their associated GrAssencharacters and "L"-series, and applications to the construction of abelian extensions of quadratic imaginary fields. Next comes a treatment of elliptic curves over function fields and elliptic surfaces, including specialization theorems for heights and sections. This material serves as a prelude to the theory of minimal models and NA(c)ron models of elliptic curves, with a discussion of special fibers, conductors, and Ogg's formula. Next comes a brief description of "q"-models for elliptic curves over C and R, followed by Tate's theory of "q"-models for elliptic curves with non-integral "j"-invariant over "p"-adic fields. The book concludes with the construction of canonical local height functions on elliptic curves, including explicit formulas for both archimedean and non-archimedean fields.

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Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1986

Количество страниц: 525

Добавлена в каталог: 22.11.2009

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID

Предметный указатель

Riemann hypothesis for varieties overfinite fields      61 81
Riemann surface      14 19 23 27;
Riemann surface, differential 1-forms on      27 43
Riemann surface, meromorphic k-forms on      27
Riemann — Roch theorem      28 30 32 87 198 226 238 345 430
Rigid analysis      430
Rigidity lemma      296 396
Ring of integers of a quadratic imaginary field      85
Ring, Krull dimension of      301
Ring, of correspondences      70
Ring, regular local      302
Rohrlich, D.      485
Root lattice of type $E_{n}$       254
Root of unity, group scheme of      397 398
Root of unity, primitive      21 23 29 154
Rosen, M.      385 485
Ruled surface      236
S-integers, effective methods      277 288
S-integers, of a function field      275
S-integers, points on an elliptic surface      275
S-morphism      297 299
S-scheme      297
S-scheme, fiber product      299
S-scheme, morphism of      297
S-scheme, proper      301
S-scheme, set of sections      298 309 397
S-scheme, set of T-valued points      298 309 397
S-scheme, smooth      301
S-unit equation over function fields      278
Saito, T.      389 390
Scheme      434
Scheme, dimension      302
Scheme, excellent      311
Scheme, fiber product      299
Scheme, finite type      311
Scheme, flat      311
Scheme, generic fiber      300
Scheme, generic point      304
Scheme, group      306
Scheme, integral      311
Scheme, nice      311 399
Scheme, non-singular point      302
Scheme, normal      311 328
Scheme, of Dedekind domain      302
Scheme, over a base ring      297
Scheme, over a base S      297
Scheme, projective closure      371
Scheme, proper      305 321 322 329 399
Scheme, reduction modulo p      300
Scheme, regular      301 398 399
Scheme, regular of dimension one      302
Scheme, regular point      302 397
Scheme, smooth      328 397 401
Scheme, smooth part      399
Scheme, structure sheaf on      397
Schmidt, W.      277
Schneider, T.      108
Schnirelmann’s Theorem      429
Second Bernoulli polynomial      468 473 478 480
Second kind differential form      43
Section, divisor of a      245
Section, to a morphism      202
Section, to an elliptic surface      202
Section, to an S-scheme      298 309 397
Segre embedding      261
Self-intersect ion      234 238 243 283 342 349 351
Self-intersect ion, equals minus one      344 352
Semi-stable reduction      388
Separated morphism      305
Serre, J.-R      140 149 261 381 447 487
Serre’s theorem      261
Shafarevich, I.R.      317 338 344
Shatz, S.      487
Sheaf, global sections      397
Sheaf, of relative differentials      306
Shimura, G.      95
Shioda, T.      189 247 254 486
Siegel, C.L.      62 275 278
Sign of functional equation      176
Silverman, J.H.      265 266 269 271 277 385 486 487
Sine function      56 58
Singular fiber      203
Singular j-invariant      104
Singular point      471
Singular point, on fiber      314
Singular point, on special fiber      325
Smooth arithmetic surface      311 316
Smooth morphism      305 306
Smooth morphism, composition of      304 320
Smooth morphism, intuitive definition      301
Smooth morphism, of relative dimension zero      306
Smooth scheme      312 397 399
Smooth scheme, map to proper scheme      328
Smooth scheme, morphism to group scheme      327 333 337
Smooth scheme, over a DVR      304
Smooth scheme, surjectivity of reduction      330 333 337 401
Smooth scheme, Weierstrass equation      321 329 362 369 378 399 400
Smooth special fiber      401
Special fiber, dual to Dynkin diagram      353
Special fiber, Galois action on      353
Special fiber, group of components      363 364
Special fiber, identity component      326 361
Special fiber, incidence matrix      350 402 403 486
Special fiber, Kodaira notation      353
Special fiber, multiplicity one components      402
Special fiber, N $\acute{e}$ ron notation      353
Special fiber, number of components      363 364
Special fiber, of an arithmetic surface      311 314 361 399
Special fiber, of blow-up      347 371
Special fiber, of curve of genus two      355
Special fiber, of minimal proper regular model      350 352 403
Special fiber, of N $\acute{e}$ ron model      350 361
Special fiber, over a DVR      300
Special fiber, singular points      325
Special fiber, smooth      401
Special linear group      8 292;
Special linear group, action on $\mathcal{D}_{N}$       72
Special linear group, reduction mod N      86
Special orthogonal group      396
Specialization map      271
Specialization map, constructing high rank curves      272
Specialization map, for height      265 266 269 281 286
Specialization map, is a homomorphism      271
Specialization map, is injective      271 281
Specialization map, on split elliptic surface      271
Spectrum of a ring      259
Split elliptic surface      220 286
Split elliptic surface, canonical height on      281
Split elliptic surface, characterization of      220 228 280
Split elliptic surface, has constant j      221
Split elliptic surface, if infinitely many bounded height points      222
Split elliptic surface, Mordell — Weil theorem      231 281
Split elliptic surface, specialization map is injective      281
Split elliptic surface, specialization of height      281
Split multiplicative reduction      362 377 379 399
Split prime      184
Sporadic groups      61
Square-free conductor      388
squares in      442
Stabilizer      11
Stark, H.      141
Strict Henselization      337
Strict Henselization, faithfully flat      338
Strict Henselization, fraction field      332
Strict Henselization, of a DVR      331
Strict Henselization, universal mapping property      332 401
Strictly Henselian ring      330 402
Strictly Henselian ring, $\acute{e}$ tale topology      332
Strictly Henselian ring, density of reduction      330 333 337
Strictly Henselian ring, N $\acute{e}$ ron model over      332 335

Structure sheaf      397
Sum, telescoping      269
Supersingular j-invariant      104
Supersingular reduction      184
surface      see also Fibered surface Elliptic Arithmetic
Surface, adjunction formula      234
Surface, ample divisor on      258
Surface, divisor      231
Surface, exceptional curve      236
Surface, fibered      236
Surface, intersection pairing      233 238
Surface, linear equivalence of divisors      232
Surface, local ring at a curve      231
Surface, local ring at a point      231
Surface, Picard group of      232
Surface, principal divisor      232
Surface, relatively minimal      235
Surface, resolution of singularities      235
Surface, ruled      236
Swan character      405
Symmetric product      229
Symmetric product, of a curve      197 402
Symplectic group      396
Szpiro, L.      388 486
Szpiro’s conjecture      388
Szpiro’s conjecture, function field analogue      287 388
Szpiro’s conjecture, used to bound torsion      453
Tame part of the conductor      380 381 405
Tame part of the conductor, invariant under isogeny      404
Tangent line      422 — 429 472
Tate curve $E_{q}$       423 441 444;
Tate curve $E_{q}$ , $\Gamma$ -invariant      441
Tate curve $E_{q}$ , discriminant of      377 423 424 473
Tate curve $E_{q}$ , filtration of rational points      430 450
Tate curve $E_{q}$ , formal group of      431
Tate curve $E_{q}$ , formula for discriminant      411
Tate curve $E_{q}$ , formula for j      411
Tate curve $E_{q}$ , has split multiplicative reduction      377 441
Tate curve $E_{q}$ , isogeny of      451 453
Tate curve $E_{q}$ , j-invariant of      423 425 438 473
Tate curve $E_{q}$ , local height on      473
Tate curve $E_{q}$ , N $\acute{e}$ ron model      377 435
Tate curve $E_{q}$ , reduction modulo $\mathcal{M}$       377 430 442
Tate curve $E_{q}$ , scheme associated to      377 434
Tate curve $E_{q}$ , surjectivity of p-adic uniformization      429 — 438
Tate curve $E_{q}$ , torsion subgroup      445
Tate curve $E_{q}$ , uniformization compatible with Galois      424
Tate curve $E_{q}$ , uniformization over p-adic fields      141 377 423 450 473 480
Tate curve $E_{q}$ , Weierstrass equation      423 426 442 473
Tate curve $E_{q}$ , Weil — Ch $\hat{a}$ telet group      452
Tate module      380
Tate module, action of Frobenius      172
Tate module, of a CM elliptic curve      148 149 179
Tate module, of a field      452
Tate module, of an abelian group      382
Tate module, of an elliptic curve      452
Tate module, of the additive group      382
Tate module, of the multiplicative group      382
Tate module, p-adic      186
Tate module, Weil pairing      125
Tate, J.      140 149 173 265 269 364 381 441 454 455 459 486 487
Tate’s algorithm      353 361 364 387 389 435
Tate’s algorithm, minimal Weierstrass equation      364
Tate’s algorithm, multiplicative reduction      377
Tate’s algorithm, verification      369 — 377
Tate’s p-adic uniformization theorem      377 441
Tchebotarev density theorem      161
Telescoping sum      269 460
Theorem 90 of Hilbert      193
Theta function $\theta$ (u, q)      412 429
Theta function $\theta$ (u, q), functional equation      412 429 474
Theta function $\theta$ (u, q), p-adic      429 473
Theta function $\theta$ (u, q), relation to Weierstrass $\sigma$ function      412
Theta function $\theta$ (u, q), relation to X(u,q) and Y(u,q)      412 429 474
Third kind differential form      43
Topological group      85 119
Topological space, irreducible      280
Topology, $\acute{e}$ tale      332
Topology, on a valued field      455
Topology, on an elliptic curve      455
Topology, pro-finite      120 182
Torsion module is sum of p-primary components      157 170
Torsion point, boundedness conjecture      453
Torsion point, field generated by      128 135 383
Torsion point, forces good/multiplicative reduction      453
Torsion point, of multiplicative group      151
Torsion point, of order eleven      190 279
Torsion point, of order five      278
Torsion point, of order seven      278
Torsion point, on CM curve form free module      102 138
Torsion point, on CM elliptic curves      102 135
Torsion point, on elliptic curves over p-adic fields      383 445
Torsion point, reduction is injective on      162
Torsion subgroup of Picard group      192 195 199
Totally imaginary field      116
Transcendence of j      108
Translation map      245 248 251
Translation map, on a group scheme      336
Translation map, on an elliptic surface      245 248 251
Translation map, on elliptic curve      310
Translation map, on group scheme      310 337 400
Translation map, on N $\acute{e}$ ron model      400
Transvection      445 447
Transversal intersection      232 282
Triangle inequality      459 471
Triplication formula, for local height      463
Twist of a cusp form      94
Twist of an elliptic curve      414 439
Twist of an L series      94
Ueno, K.      355 486
Uniformization theorem      6 34 35 88 96 196 411 413
Uniformization Theorem, of elliptic curves over $\mathbb{R}$       416
Uniformization Theorem, of elliptic curves over p-adic fields      423 441
Uniformizer      27
Uniformizer, for a curve on an arithmetic surface      339
Unit disk, holomorphic automorphism of      22
Unit group of a function field      191
Unit group of a number field      191
Universal family, point of order eleven      190 279
Universal property of fiber product      299
Universally closed morphism      305
Unramified extension, cyclotomic      383
Unramified extension, generated by torsion      383
Unramified extension, of discrete valuation ring      399
Unramified Gr $\ddot{o}$ ssencharacter      168 173 174
Upper half plane H      7
Upper half plane H, action of $SL_{2}(\mathbb{Z})$       9
Upper half plane H, differential form on      26
Upper half plane H, extended (H*)      see Extended upper half plane
Upper half plane H, fundamental domain for $SL_{2}(\mathbb{Z})$ action      10 14 25 92
Upper half plane H, stablizer of a point      11
v-adic topology, multiplication map is continuous      460
v-adic topology, on a field      455
v-adic topology, on an elliptic curve      455 476
Valuation of a curve on an arithmetic surface      311 313 339
Valuation on a function field      194
Valuative criterion of properness      303 317 329 333
Variety, abelian      196
Variety, birational equivalence of      204 205
Variety, category up to birational equivalence      205
Variety, curve through two points      228 281
Variety, Frobenius map of      132
Variety, group      291
Variety, group of divisors      255
Variety, height associated to a morphism      258
Variety, height on      256
Variety, rational map of      132
Vertical strip      94
Very ample divisor      257 261 284
Vojta, P.      278

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