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

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Предметный указатель

Elliptic curve, minimal discriminant      363 364
Elliptic curve, minimal proper regular model      325 332 335 337 350 352 361 400
Elliptic curve, multiplication map      310
Elliptic curve, multiplicative reduction      403
Elliptic curve, N $\acute{e}$ ron model      319
Elliptic curve, negation map      325
Elliptic curve, normalized      97 131 134
Elliptic curve, of high rank      272
Elliptic curve, ordinary reduction      184
Elliptic curve, over $\mathbb{R}$       see Elliptic curves over $\mathbb{R}$
Elliptic curve, over C      see Elliptic curves over C
Elliptic curve, over function field      see Elliptic surface
Elliptic curve, over locally compact field      476
Elliptic curve, over p-adic fields      see Elliptic curves over p-adic fields
Elliptic curve, potential good reduction      140 447
Elliptic curve, reduction modulo $\beta$       131
Elliptic curve, reduction type      352 353 363 364
Elliptic curve, representation on m-torsion      108 383
Elliptic curve, supersingular reduction      184
Elliptic curve, translation map      310
Elliptic curve, twists of      283 414 439
Elliptic curve, v-adic topology      see v-adic topology on
Elliptic curve, Weber function      134 135
Elliptic curve, with complex multiplication      see Complex multiplication
Elliptic curves over $\mathbb{C}$       408 — 413
Elliptic curves over $\mathbb{C}$ , analytic parametrization      6 34 408 411 464 468
Elliptic curves over $\mathbb{C}$ , discriminant      26 409 411 464 467;see
Elliptic curves over $\mathbb{C}$ , homology group      37 43
Elliptic curves over $\mathbb{C}$ , isomorphism class of      36 37
Elliptic curves over $\mathbb{C}$ , j-invariant      34 36 411;
Elliptic curves over $\mathbb{C}$ , lattice of      408 468 477
Elliptic curves over $\mathbb{C}$ , local height      464 466 468 477
Elliptic curves over $\mathbb{C}$ , periods      37 43
Elliptic curves over $\mathbb{C}$ , Uniformization Theorem      6 34 35 88 96 196 411
Elliptic curves over $\mathbb{C}$ , Weierstrass equation      34 409 464
Elliptic curves over $\mathbb{R}$       413 — 422
Elliptic curves over $\mathbb{R}$ , $\Gamma$ -invariant      414 419 450
Elliptic curves over $\mathbb{R}$ , $\Gamma$ -invariant, not equal one      450
Elliptic curves over $\mathbb{R}$ , $\mathbb{C}$ isomorphism classes      414
Elliptic curves over $\mathbb{R}$ , $\mathbb{R}$ isomorphism classes      414 416
Elliptic curves over $\mathbb{R}$ , homogeneous spaces      449
Elliptic curves over $\mathbb{R}$ , isomorphic to real Lie group      420
Elliptic curves over $\mathbb{R}$ , Kummer sequence      449
Elliptic curves over $\mathbb{R}$ , number of components      419
Elliptic curves over $\mathbb{R}$ , sign of the discriminant      420 421 449
Elliptic curves over $\mathbb{R}$ , torsion subgroup      449
Elliptic curves over $\mathbb{R}$ , twists      414 450
Elliptic curves over $\mathbb{R}$ , uniformization over $\mathbb{R}$       416
Elliptic curves over $\mathbb{R}$ , Weil — Ch $\hat{a}$ telet group      421 449
Elliptic curves over $\mathbb{R}$ , with CM      142 179
Elliptic curves over non-archimedean fields      424
Elliptic curves over p-adic fields      438 — 444; see also Tate curve $E_{q}$
Elliptic curves over p-adic fields, $\Gamma$ -invariant      439 441 451
Elliptic curves over p-adic fields, $\Gamma$ -invariant, not equal one      444
Elliptic curves over p-adic fields, additive reduction      451
Elliptic curves over p-adic fields, applications      445 — 448
Elliptic curves over p-adic fields, local height      469 472 480
Elliptic curves over p-adic fields, multiplicative reduction      377 478
Elliptic curves over p-adic fields, non-split multiplicative reduction      378 451
Elliptic curves over p-adic fields, quadratic character      440 444
Elliptic curves over p-adic fields, split multiplicative reduction      377 441 451 452 472 480
Elliptic curves over p-adic fields, Tate module of      452
Elliptic curves over p-adic fields, torsion points      383 445
Elliptic curves over p-adic fields, uniformization      141 423 441 444 473 480
Elliptic curves over p-adic fields, Weil — Ch $\hat{a}$ telet group      452
Elliptic function      39; see also Weierstrass $\Gamma$ function Weierstrass Weierstrass
Elliptic function, divisor of      45
Elliptic function, factorization as product of $\sigma$ ’s      45
Elliptic function, Laurent series of F(u; q)      49
Elliptic integral      178
Elliptic surface      see also Fibered surface Surface
Elliptic surface, $ord_{t}(x)$ is bounded      275
Elliptic surface, algebraic equivalence      254
Elliptic surface, as subset of $\mathbb{P}^{1}\times C$       200
Elliptic surface, associated elliptic curve      206
Elliptic surface, automorphism      245
Elliptic surface, bad fiber      203
Elliptic surface, birational equivalence      206
Elliptic surface, canonical divisor      249
Elliptic surface, canonical height on      217 247 252 265 266 269 281 284 286
Elliptic surface, classical theory      187
Elliptic surface, conductor divisor      287
Elliptic surface, definition of      202 203 205
Elliptic surface, divisor $D_{P}$       247
Elliptic surface, divisor of a section      245
Elliptic surface, dominant rational map      208 222 228
Elliptic surface, fiber      201 202
Elliptic surface, fiber product      211 301
Elliptic surface, field of definition      203
Elliptic surface, field of definition, of section      189 282
Elliptic surface, generic fiber      206
Elliptic surface, good fiber      203
Elliptic surface, group of sections      202 210 298
Elliptic surface, group of sections, defined over k      203 210
Elliptic surface, height on      212 213 265
Elliptic surface, infinitely many points of bounded height      220 222
Elliptic surface, is a C-scheme      298
Elliptic surface, j-invariant, constant      286
Elliptic surface, j-invariant, is algebraic function      280
Elliptic surface, Kummer sequence for      193
Elliptic surface, lattice structure      254
Elliptic surface, lower bound for height      287
Elliptic surface, map on Pic is injective      253
Elliptic surface, minimal discriminant divisor      286
Elliptic surface, Mordell — Weil theorem      230 276 279
Elliptic surface, multiple fibers      203
Elliptic surface, multiplication-by-n map      266
Elliptic surface, N $\acute{e}$ ronl — Severi group      254
Elliptic surface, non-singular fiber      286
Elliptic surface, non-split      218 247
Elliptic surface, non-split, with zero discriminant      286
Elliptic surface, one-cocycle on sections      284
Elliptic surface, one-parameter family      188
Elliptic surface, over non-perfect field      187
Elliptic surface, points not a group      211
Elliptic surface, rational map between      206
Elliptic surface, S-integral points on      275 277 288
Elliptic surface, section      202
Elliptic surface, section, of order eleven      190 279
Elliptic surface, section, of order five      278
Elliptic surface, section, of order seven      278
Elliptic surface, singular fiber      203
Elliptic surface, specialization map      271
Elliptic surface, specialization map, is injective      271 281
Elliptic surface, specialization map, of height      265 266 269 281 286
Elliptic surface, split      220 222 228 231 271 280 281
Elliptic surface, split over $\mathbb{P}^{1}$       286
Elliptic surface, subgroup $E_{0}(K)$       251 284 288
Elliptic surface, torsion subgroup      288
Elliptic surface, weak Mordell — Weil theorem      191 — 195 230
Elliptic surface, zero section      202
Endomorphism ring, commutator subring      129
Endomorphism ring, elliptic curve with given      121
Endomorphism ring, normalized isomorphism      97 129 131 134
Endomorphism ring, quaternion algebra      130
Endomorphism ring, reduction modulo $\beta$       129
Endomorphism, field of definition of      106
Endomorphism, of an elliptic curve      448
Endomorphism, of degree two      109 141 179
Eta function ( $\eta$ )      see Quasi-period homomorphism Dedekind
Euclidean structure on Mordell — Weil group      247
Euler product      80 92 172 173
Excellent scheme      311
Exceptional curve      236 244
Exceptional divisor      344
Exceptional divisor, Castelnuovo’s criterion      344 352
Exceptional divisor, on an arithmetic surface      344
Exponent of the conductor      380 389
Exponent of the conductor, bound for      385
Exponent of the conductor, computation of      389 407

Exponent of the conductor, independent of l      381 405
Exponent of the conductor, less than discriminant      396
Exponential function, special values      151
Exponential map on multiplicative group      151
Extended upper half plane H*      14
Extended upper half plane H*, action of $\Gamma$ (1)      14 85
Extended upper half plane H*, cusps of      14 85
Extended upper half plane H*, is a Hausdorff space      16
Extended upper half plane H*, topology on      16 85
Faithfully flat, descent      338
Faithfully flat, strict Henselization is      338
Faltings, G.      271
Fermigier, T.      272
Fiber product      299
Fiber product, diagonal      300 309 327
Fiber product, graph in      300
Fiber product, group operation on      211
Fiber product, of elliptic surface      211 301
Fiber product, universal property      299
Fiber, archimedean      345
Fiber, closed      300
Fiber, generic      300
Fiber, multiplicity of components      313
Fiber, of a scheme over a point      300
Fiber, of an arithmetic surface      313
Fiber, of an elliptic surface      201
Fiber, singular point      314
Fiber, smooth      397
Fiber, special      300
Fibered surface      236; see also Surface Elliptic
Fibered surface, connected fibers      240 283 342
Fibered surface, fiber of      236
Fibered surface, fiber of, genus at least one      244
Fibered surface, fibral curve      237
Fibered surface, fibral divisor      237
Fibered surface, fibral homomorphism      237
Fibered surface, fibral intersections      238
Fibered surface, horizontal curve      237
Fibered surface, horizontal divisor      237
Fibered surface, map on Picard group      284
Fibered surface, map on Picard group, is injective      253
Fibered surface, minimal      244
Fibered surface, multiplicity of fibers      237
Fibered surface, negative semi-definite intersection pairing      238
Fibered surface, relatively minimal      243
Fibral curve      237
Fibral divisor      237 246 247 284
Fibral divisor, group of      340 344
Fibral divisor, intersects fiber trivially      238
Fibral divisor, on an arithmetic surface      340
Fibral divisor, self-intersection      238 342 351
Field of definition, for E( $\mathbb{C}$ T))      282
Field of definition, of an abelian variety      196
Field of definition, of an elliptic curve      37 38
Field of definition, of an elliptic surface      203
Field of definition, of Jacobian variety      199
Field of definition, of section to elliptic surface      189 282
Field of moduli      37 38
Field of moduli of an elliptic curve      37 38
Field, locally compact      476
Field, non-perfect      187
Field, of characteristic zero      187 189
Field, perfect      330 332
Field, totally imaginary      116
Finite group scheme      397
Finite order elements in Picard group      192 195 199
Finite type scheme      311
First kind differential form      43
Fischer — Griess monster group      61
Flat family, arithmetic genus in      343
Flat morphism      304 397
Flat scheme      311 345
Formal Dirichlet series      80
Formal group of an elliptic curve      276 381
Formal group of Tate curve $E_{q}$       431
Formal logarithm      150 182
Formal multiplicative group      149 431
Fourier analysis      173
Fourier coefficients      see also q-expansion
Fourier coefficients of a simultaneous eigenfunction      79 80
Fourier coefficients, action of Hecke operator      76 78 79
Fourier coefficients, integer      145
Fourier coefficients, relations satisfied by      79 80
Fourier coefficients, size of      81 82 92
Fourier expansion      see q-expansion
Fractional ideal is a lattice      99
Fractional ideal of quadratic imaginary field      85
Free module of rank one      102 138 186
Frey, G.      453
Fricke, M.      95
Frobenius map, action on Tate module      172
Frobenius map, automorphism      116 128 129 130 132 161
Frobenius map, characteristic polynomial      172
Frobenius map, lift to characteristic zero      130 132 162
Frobenius map, of varieties      132
Fudge factor      364
Function field, birational equivalence      205
Function field, canonical height is rational      219 247
Function field, elliptic curve over      see Elliptic surface
Function field, finite subgroup of      195
Function field, height on      212
Function field, height on, elliptic curve over      212 213
Function field, ideal class group      192
Function field, infinitely many elements of bounded height      213
Function field, Kummer theory      191
Function field, local heights      212
Function field, map induced by dominant rational map      205
Function field, Mordell — Weil theorem      230 279
Function field, S-integers      275
Function field, Szpiro’s conjecture      287 388
Function field, unit group of      191
Function field, valuation on      194
Functional equation, for $\theta$ (u,g)      412 429 474
Functional equation, for X(u, q)      425
Functional equation, for Y(u, q)      425
Functional equation, of Dirichlet series attached to cusp form      94
Functional equation, of L-series      176 184
Functional equation, of L-series attached to cusp form      83 94
Functional equation, sign of      176
Functor, contravariant      299 309
Functor, defined by a group scheme      309
Functor, defined by an S-scheme      298 306
Functor, Jacobian      198
Functoriality of height function      256
Fundamental domain ( $\mathcal{F}$ ) for action of $SL_{2}(\mathbb{Z})$ on H      10 14 82 92
Fundamental domain, for $\mathbb{C}/\Lambda$       41 71
Galois action on series      424 428 451
Galois action on special fiber      353
Galois cohomology of a cyclic group      421
Galois cohomology of a discrete group      428
Galois conjugate of a CM elliptic curve      131
Galois group, decomposition group      331
Galois group, higher ramification group      379
Galois group, inertia group      120 149 331
Galois group, of Hilbert class field      118
Galois theory      37
Gamma function      83 176
Gamma function, incomplete      93
Gauss sum      93
Gauss, K.F.      141
Gaussian integers      178
Gel’fond — Schneider theorem      108 142
General linear group      291
General linear group, centralizer of a subgroup      179
General linear group, determinant      292
General linear group, elements of finite order      403 404
General linear group, formal logarithm      150 182
General linear group, rational points of      292
Generic fiber      300
Generic fiber, empty      401

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