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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.


Язык: en

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
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Предметный указатель
Multiplicative reduction, non-split      171 378
Multiplicative reduction, split      171 362 377 379
Multiplicative reduction, torsion for      453
Multiplicity of fibral components      313
N$\acute{e}$ron height function      see Local height function
N$\acute{e}$ron mapping property      319 329 333 338
N$\acute{e}$ron model      319 361
N$\acute{e}$ron model, addition formula on Weierstrass equation      323
N$\acute{e}$ron model, automorphism of      400
N$\acute{e}$ron model, connected component      326 361 401
N$\acute{e}$ron model, dual isogeny      400
N$\acute{e}$ron model, existence      325 332 335 337
N$\acute{e}$ron model, extension of group law      332 335
N$\acute{e}$ron model, generic fiber      319
N$\acute{e}$ron model, group of components      350 362 363 364 379 402 486
N$\acute{e}$ron model, group of sections      319
N$\acute{e}$ron model, group scheme action      326 400
N$\acute{e}$ron model, identity component is Weierstrass equation      362 378
N$\acute{e}$ron model, identity component of special fiber      326 361
N$\acute{e}$ron model, identity section      326
N$\acute{e}$ron model, isogeny      400
N$\acute{e}$ron model, multiplicative reduction      403
N$\acute{e}$ron model, N$\acute{e}$ron mapping property      319 320
N$\acute{e}$ron model, negation map on Weierstrass equation      325
N$\acute{e}$ron model, not proper      319
N$\acute{e}$ron model, of Jacobian variety      350 486
N$\acute{e}$ron model, of Tate curve $E_{q}$      435
N$\acute{e}$ron model, over a strictly Henselian ring      402
N$\acute{e}$ron model, over strictly Henselian ring      332 335
N$\acute{e}$ron model, special fiber      350 361 400 435
N$\acute{e}$ron model, subgroup scheme      401
N$\acute{e}$ron model, translation map      400
N$\acute{e}$ron model, uniqueness      319
N$\acute{e}$ron model, unramified base extension      320
N$\acute{e}$ron model, Weierstrass equation      321 329 362 369 378 399 400
N$\acute{e}$ron notation for special fiber      353
N$\acute{e}$ron — Kodaira classification of fibers      352
N$\acute{e}$ron — Ogg — Shafarevich criterion      140 151 148 170 362 381 383
N$\acute{e}$ron — Severi group of an elliptic surface      254
N$\acute{e}$ron — Severi group, rank      254
N$\acute{e}$ron, A.      231 265 272 352 454 455 485
Nagao, K.      272
Nakai — Moishezon criterion      258 261 283 284
Nakayama’s Lemma      103
Namikawa, Y.      355 486
Natural map      197 198
Negation map, on an elliptic curve      325
Nice scheme      311 399
Node      287 370
Non-linearity, textual      217
Non-perfect field      187
Non-singular arithmetic surface      311
Non-singular point, on fiber of arithmetic surface      315 351 399
Non-singular point, on reduction      362 379
Non-singular point, on scheme      see Regular point
Non-singular scheme      see Regular scheme
Non-split multiplicative reduction      378 399
Norm map, compatibility with reciprocity map      120
Norm map, of fields      444
Norm map, on ideles      119
Normal law      334 335
Normal law, associativity      334 335
Normal law, induces group law      334 337
Normal law, on symmetric product of a curve      402
Normal scheme      311
Normal scheme, principal divisor on      328
Normalized basis      7
Normalized eigenfunction      78 79 80 92 93;
Normalized elliptic curve (E, [$\cdot)      97 131 134
Normalized lattice      47
Nullstellensatz      259
Number field, $\mathbb{Q}$(i)      138
Number field, $\mathbb{Z}_{p}$-extension      186
Number field, abelian extension      128; see also Class field theory
Number field, Artin map      117 118 154
Number field, decomposition group      116
Number field, ideal class group      99 180
Number field, idele group      119
Number field, maximal abelian extension      120 121
Number field, of class number one      138 141
Number field, unit group of      191
Numerical equivalence      486
of a surface at a point      231
Ogg, A.      355 389 487
Ogg’s formula      363 364 387 389 396 407
One-cocycle on elliptic surface      284
One-parameter family of elliptic curves      188 200
Operator, Hecke      see Hecke operator
Operator, homothety      68 90
Order, at $\infty$ of a modular function      24
Order, conductor of      178
Order, non-maximal      160 178 180
Order, of a differential form      27
Ordinary reduction      179 184
Oriented basis      6 9 71 72 73 89
Orthogonal group      396
Orthogonal group, special      396
Osterl$\acute{e}$, J.      278
p-adic analysis      424 429
p-Adic field      423
p-adic period      364
p-adic Tate module      186
p-primary component      157
Parallelogram law      218
Parallelogram law, for height      213
Path independent integral      198
Perfect field      330 332
Period map      43
Period, p-adic      364
Periodic second Bernoulli polynomial      480
periods      37
Petersson inner product      92
Petersson inner product, Hecke operators are self-adjoint      92
Petersson, H.      61 80
Picard group      see also Jacobian variety
Picard group, divisor class of degree zero      197
Picard group, elements of finite order      192 195 199
Picard group, induced homomorphism      255
Picard group, is ideal class group      192
Picard group, of $\mathbb{P}^{2}$      232 282
Picard group, of $\mathbb{P}^{n}$      282
Picard group, of a curve      402
Picard group, of a fibered surface      284
Picard group, of a surface      232
Picard group, of curve of genus two      287
Picard group, of hyperelliptic curve      287
Points of a group scheme      309 397
Points of a scheme      298 309 397
Polar divisor on normal scheme      328
Positive divisor      341
Positive divisor, associated height      285
Potential good reduction      140 148 151 176 383 447
Power divisor function $\sigma_{k}$      see $\sigma_{k}$
Prime ideal, in arithmetic progression      118 123
Prime ideal, in ideal class      118 123
Prime ideal, inert      184
Prime ideal, split      184
Prime ideal, splits completely      117 118 179
Primitive Dirichlet character      93
Primitive root of unity      21 23 29 154
Principal branch of logarithm      44 54 56 66
Principal branch of square root      65 67
Principal divisor      232
Principal divisor, on an arithmetic surface      339
Principal ideal      132
Principal ideal, congruent to 1 modulo c      117
Pro-finite topology      120 182
Pro-p group      150 182
Product expansion of sine function      56 58
Product, fiber      299
Product, restricted      119
Projection morphism on fiber product      299
Projective closure of a scheme      371
Projective line, arithmetic genus      343 352
Projective line, arithmetic surface      312
Projective line, proper over R      312
Projective line, smooth over R      312
Projective plane, Picard group of      232 282
Projective scheme, is proper      303 322
Projective space, height on      255
Projective space, over a ring      298
Projective space, over a scheme      301 303 322
Projective space, Picard group of      282
Proper arithmetic surface      311 316 317 351
Proper group scheme      327
Proper morphism      297 305
Proper morphism, intuitive definition      301
Proper morphism, valuative criterion      303 317 329 333
Proper regular model, minimal      325 332 335 337 389 400
Proper regular model, of a curve      317
Proper S-scheme      301 305
Proper scheme      312 321 329 399
Proper scheme, group      327
Proper scheme, map from smooth scheme      328
Proper scheme, projective scheme is      303 322
q-expansion      see also Fourier coefficients
q-expansion, of $G_{2k}$      55
q-expansion, of $g_{2}$ and $g_{3}$      409
q-expansion, of $\varphi$ and $\varphi$’      47 50 409
q-expansion, of a modular function      24 55 182
q-expansion, of Bernoulli polynomial      480
q-expansion, of discriminant $\Delta$      59 409 467 468
q-expansion, of j-invariant      59 144 145 146 410
q-expansion, of Weierstrass $\sigma$ function      53 467
q-expansion, of Weierstrass $\zeta$ function      52
Quadratic character      440 444
Quadratic field, non-maximal order      178 180
Quadratic form, canonical height      454
Quadratic imaginary field, $\mathbb{Q}(i)$      138
Quadratic imaginary field, abelian extension of      128 135
Quadratic imaginary field, algorithm to compute ideal class group      85
Quadratic imaginary field, Hilbert class field of      95 121 122 142 166 181
Quadratic imaginary field, ideal class group      85
Quadratic imaginary field, maximal abelian extension      135
Quadratic imaginary field, of class number one      138 141
Quadratic imaginary field, ring of integers      85
Quadratic transformation      244
Quasi-parallelogram law      213 467 476
Quasi-parallelogram law, explicit O(1) estimate      280
Quasi-period homomorphism      41 43 44 52 53 65 89 412 464 465 466;
Quasi-period homomorphism, formula for      41
Quasi-periodicity of $\zeta$      40
Quaternion algebra      130
Quaternion group      404
Quotient topology      85
Quotient topology, on X(1)      18 19
R-morphism      297
R-scheme      297
R-scheme, generic fiber      300
R-scheme, reduction modulo p      300
R-scheme, set of R-valued points      298
R-valued points      298
R-valued points, of an affine scheme      298
Ramanujan $\tau$ function      59 60 482
Ramanujan $\tau$ function, growth properties      61
Ramanujan $\tau$ function, multiplicative identities      61 79
Ramanujan $\tau$ function, special values      60
Ramanujan conjecture      61 81
Ramanujan, S.      61 79
Ramification index      283
Ramification, wild      380
Rank one module      102 138 186
Rank, high over $\mathbb{Q}$      272
Rank, of N$\acute{e}$ron — Severi group      254
Rank, upper bound      279
Rational map      203
Rational map, birational isomorphism      204
Rational map, blowing-up      204 235
Rational map, domain of definition      204 279
Rational map, dominant      204 222 228
Rational map, from smooth scheme to proper scheme      328
Rational map, image      204
Rational map, image, is algebraic      204 279
Rational map, image, is irreducible      204 280
Rational map, naive definition      204
Rational map, of elliptic surfaces      206
Rational map, of varieties      132
Ray class field      117 118 129 135 161
Ray class field, conductor of      117 118 129 135
Ray class field, idelic characterization      120 162
Ray class field, modulo unit ideal      118
Ray class field, of $\mathbb{Q}(i)$      138 180
Ray class field, of $\mathbb{Q}(\sqrt{ — 3})$      180
Ray class field, primes splitting completely      118
Raynaud, M.      350 486
Real Lie group      420
Reciprocity, Artin      117
Reciprocity, Dedekind      67 90
Reciprocity, map      120 152 159 165 166 168 169 174
Reduction map, injective on torsion      162
Reduction map, surjectivity      330 333 337 401
Reduction type      352 353 363 364
Reduction type, over 3-adic field      406 407
Reduction, additive      171 287 381 388 399 403
Reduction, bad      321
Reduction, good      287 321 329 381 383 388
Reduction, multiplicative      171 287 369 381 388 403
Reduction, non-singular points      362 379
Reduction, non-split multiplicative      399
Reduction, of a scheme      300
Reduction, of an isogeny      124
Reduction, of Weierstrass equation      362 378
Reduction, ordinary      179
Reduction, semi-stable      388
Reduction, split multiplicative      399
Regular arithmetic surface      316 317 351 370 399
Regular arithmetic surface, non-singular point on fiber      315 351 399
Regular arithmetic surface, smooth part      316 318 321 325 332 335 361 362 369 378 399 400
Regular arithmetic surface, Weierstrass equation      399
Regular local ring      302 370
Regular local ring, localization is regular      397
Regular local ring, of dimension one      328
Regular local ring, of dimension two      315
Regular model, minimal proper      325 332 335 337 400
Regular point      302
Regular point, on a scheme      397
Regular scheme      302 398 399
Regular scheme, example      313
Regular scheme, intuitive definition      301
Regular scheme, of dimension one      302
Regular scheme, smooth part      399
Regular scheme, Weierstrass equation      399
Regular, arithmetic surface      311
Regular, differential form      27
Regular, in codimension one      311
Regulator, height      273
Reimann conditions      196 198
Relative differentials      306
Relative dimension of a smooth mor¬phism      305
Relatively minimal fibered surface      243
Relatively minimal surface      235
Representation, irreducible      405
Representation, of type $E_{8}$      189
Residue symbol, 6th-power      177
Resolution of singularities      235
Restricted product      119
Resultant      458
Riemann $\zeta$ function      25 55 171
Riemann $\zeta$ function, functional equation      83
Riemann $\zeta$ function, special values      26 33 57
1 2 3 4 5 6 7 8
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