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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

$E_{q}$       see Tate curve $E_{q}$
$GL_{n}$       see General linear group
$G_{2k}$       see Eisenstein series
$S*_{n}$       181
$S*_{n}$ , order of      181
$SL_{2}$       see Special linear group
$s_{1}$ (q)      411 425
$s_{k}$ (q)      410
$s_{k}$ (q), alternative form      448
$S_{n}$       72 143
$S_{n}$ , order of      72
$\acute{E}$ tale morphism      306
$\acute{E}$ tale topology      332
$\Gamma$ (1)      see Modular group $\Gamma$ (1)$
$\Gamma$ -invariant, in characteristic two and three      451
$\Gamma$ -invariant, of an elliptic curve      439
$\Gamma$ -invariant, of an elliptic curve, defined over $\mathbb{R}$       414 419
$\Gamma$ _(N)$      86
$\Gamma$ _(s)$      see Gamma function
$\Gamma$ _{0}(N)$      87
$\Gamma$ _{1}(N)$      87
$\mathbb{Z}_{p}$ -extension      186
$\mathcal{D*}_{n}$       181
$\mathcal{D}_{n}$       72 91 143
$\mathcal{D}_{n}$ , action of $SL_{2}(\mathbb{Z})$       72
$\Sigma$       see Weierstrass $\sigma$ function
$\sigma_{k}$       55 61
$\tau(n)$       see Ramanujan $\tau$ function
$\varphi$       see Weierstrass $\varphi$ function
$\zeta$ function      see also Weierstrass $\zeta$ function Riemann
$\zeta$ function of a number field      183
$\zeta$ function of an elliptic curve over a finite field      183
$\zeta$ function of an elliptic curve over a number field      183
ABC-Conjecture      278
Abel — Jacobi theorem      198
Abelian extensions      see also Class field theory
Abelian extensions of $\mathbb{Q}$       129 151
Abelian extensions of Hilbert class field      134
Abelian extensions, Artin map      117
Abelian extensions, conductor      117 118 123
Abelian extensions, Frobenius element $\sigma_{p}$       116 121 128 129 130 132
Abelian extensions, generated by       121
Abelian extensions, generated by CM torsion points      108 128 135 148 149 166 168
Abelian extensions, generated by roots of unity      108 128 153
Abelian extensions, inertia group      120 149
Abelian extensions, maximal      120 121
Abelian extensions, ray class field      117 118 120 129 135 180
Abelian extensions, unramified      see Hilbert class field
Abelian group, bilinear form on      273
Abelian group, is sum of p-primary components      152 157
Abelian variety      148 196
Abelian variety, automatically commutative      196
Abelian variety, field of definition      196
Abelian variety, Jacobian      197 402
Abelian variety, Mordell — Weil theorem      485
Abelian variety, of dimension one      196 279
Abelian variety, uniformization theorem over $\mathbb{C}$       196 199
Abelian variety, with complex multiplication      96
Abhyankar, S.S.      317
Absolute value defines a topology      455
Absolute value on a field      455
Absolute value on a number field      461
Action of $SL_{2}(\mathbb{Z})$ on $\mathcal{D}_{n}$       72
Action of $SL_{2}(\mathbb{Z})$ on $\mathfrak{H}$       9
Action of $SL_{2}(\mathbb{Z})$ on $\mathfrak{H}$ , fundamental domain      10 92
Action of $SL_{2}(\mathbb{Z})$ on $\mathfrak{H}$ , stabilizer of a point      11
Action of $SL_{2}(\mathbb{Z})$ on $\mathfrak{H}*$       14
Action of a Hecke operator on a cusp form      76 78 79
Action of a Hecke operator on a lattice function      74
Action of a Hecke operator on a modular form      76
Action of a Hecke operator on a modular function      76 79 91
Action of a Hecke operator on Fourier coefficients      76 78
Action, group scheme      321 326 400
Addition formula on an elliptic curve      210 216 323 471
Additive group      398 399
Additive group, over $\mathbb{Z}$       307
Additive group, over a field      291 293 398
Additive group, over R      307
Additive group, over S      307
Additive group, rational points of      292
Additive group, S-valued points      397
Additive group, Tate module of      382
Additive reduction      171 287 388 399 403
Additive reduction, conductor      381
Additive reduction, torsion limited      453
Adjunction formula      234 249 345
Adjunction formula, on arithmetic surface      342 351
Affine group variety      396
Affine scheme, set of R-valued points      298
Affine space over a ring      298
Affine space over a scheme      301
Algebraic equivalence      486
Algebraic equivalence of divisors      254 285
Algebraic equivalence of points      201
Algebraic family of elliptic curves      201
Algebraic group      see Group variety
Algebraic group variety      115
Algebraic integer, j-invariant is      140 147 151 447
Algorithm, Laska’s      364
Algorithm, minimal Weierstrass equation      364 403
Algorithm, Tate’s      353 361 364 387 389 435
Algorithm, to compute local heights      478 479
Almost every      201
Almost quadratic function      454 455;
Ample divisor      233 257
Ample divisor, height associated to      257
Ample divisor, on a curve      257 264
Ample divisor, on a surface      258
Ample divisor, Serre’s theorem      261
Analytic continuation of L-series attached to cusp form      83 94
Approximation theorem, weak      155 158
Arakelov intersection theory      344
Arakelov intersection theory, adjunction formula      345
Arakelov intersection theory, Riemann — Roch theorem      345
Arakelov, S.      344
Archimedean fiber      345
Archimedean local height      464
Arithmetic genus      342 351
Arithmetic genus, in flat family      343
Arithmetic genus, one      355
Arithmetic genus, zero      343 352
Arithmetic surface      311
Arithmetic surface, adjunction formula      342 351
Arithmetic surface, Arakelov intersection theory      344
Arithmetic surface, arithmetic genus of curve on      342 351
Arithmetic surface, blow-up      343 345 371
Arithmetic surface, blow-up, example      347 402
Arithmetic surface, blow-up, of a curve      348 372
Arithmetic surface, canonical divisor      342 350 351
Arithmetic surface, closed fiber      313
Arithmetic surface, connected fiber      342 353
Arithmetic surface, dual graph of special fiber      353
Arithmetic surface, exceptional divisor      344
Arithmetic surface, extension of automorphisms      318 336
Arithmetic surface, flat      345
Arithmetic surface, Galois action on special fiber      353
Arithmetic surface, generic fiber      311
Arithmetic surface, incidence matrix      350 402 403 486
Arithmetic surface, intersection pairing      341 342 353
Arithmetic surface, intuitive definition      311
Arithmetic surface, is curve over R      311
Arithmetic surface, local intersection index      339
Arithmetic surface, local ring of a closed point      315
Arithmetic surface, local ring of a curve      311 313 339
Arithmetic surface, local ring of a point      370
Arithmetic surface, minimal proper regular model      317 318 344 361
Arithmetic surface, model of curve of genus two      355
Arithmetic surface, negative semi-definite intersection pairing      342 353
Arithmetic surface, non-singular      311
Arithmetic surface, non-singular point on fiber      315 351 399

Arithmetic surface, projective line $\mathbb{P}^{1}_{R}$       312
Arithmetic surface, proper      311 316 317 351
Arithmetic surface, proper regular model      317
Arithmetic surface, regular      311 315 316 317 351 370 399
Arithmetic surface, regular in codimension one      311
Arithmetic surface, resolution of singularities      317
Arithmetic surface, singular point on fiber      314
Arithmetic surface, smooth      311
Arithmetic surface, smooth part      316 318 321 325 332 335 361 362 369 378 399 400
Arithmetic surface, special fiber      311 314 350 352 361 371 399
Arithmetic surface, special fiber of blow-up      347 371
Arithmetic surface, Weil divisor on      311
Artin character      405
Artin map      117 118 120 174
Artin map for $\mathbb{Q}$       154
Artin map for Hilbert class field      118
Artin map, is surjective      118
Artin map, kernel of      117 118 179
Artin map, prime splits completely      117 179
Artin map, trivial on norms      179
Artin reciprocity      117
Artin, M.      325 327 334
Associative law, of a group scheme      307
Associativity, normal law      334 335
Automorphism group of an elliptic curve      183
Automorphism group of an elliptic surface      245
Automorphism of a curve      294
Automorphism of arithmetic surface      318 336
Automorphism of N $\acute{e}$ ron model      400
Bad fiber      203
Bad reduction      185 321
Baker, A.      141
Basis for a lattice, normalized      7
Basis for a lattice, oriented      6 9 71 72 73 89
Bernoulli numbers      57
Bernoulli polynomial, Fourier expansion      480
Bernoulli polynomial, second      468 473 478 480
Bertini’s theorem      233
Bezout’s Theorem      232 233
Bilinear form, non-degenerate      273
Birational equivalence      204 205
Birational equivalence of elliptic surfaces      206
Birational equivalence over $\mathfrak{k}$       204
Birational equivalence, category of elliptic surfaces up to      206
Birational equivalence, category of varieties up to      205
Birational isomorphism      204
Birational map of fibered surface      244
Birational morphism      235
Birch — Swinnerton — Dyer fudge factor      364
Birch, B.J.      389
Blanksby, P.      487
Blow-down      236 244
Blow-down, Castelnuovo’s criterion      344
Blow-up      343
Blow-up of a scheme      434
Blow-up of a Weierstrass equation      371
Blow-up of multiplicative reduction curve      403
Blow-up, Castelnuovo’s criterion      344
Blow-up, coordinate chart      345 347 371
Blow-up, example      347 402
Blow-up, exceptional divisor      344
Blow-up, gluing of charts      346 348 371
Blow-up, height associated to      285
Blow-up, of an arithmetic surface      343 345
Blow-up, primer on      345
Blow-up, special fiber      347 348 371
Bounded conductor      404
Boundedness conjecture      453
Brauer group, of a local field      452
Brumer, A.      385
Calculus, freshman      50
Canonical divisor      27 87
Canonical divisor on an arithmetic surface      342 350 351
Canonical divisor on an elliptic surface      249
Canonical height      252 454
Canonical height pairing      218 247 252 265 269 286
Canonical height pairing, non-degeneracy      272
Canonical height pairing, with values in Pic()      252 265
Canonical height pairing, with values in Pic( $\varepsilon$ )      284
Canonical height, as intersection index      247
Canonical height, as limit      454
Canonical height, as sum of local heights      454 461
Canonical height, associated to a divisor      285
Canonical height, comparison with naive height      478
Canonical height, explicit (1) estimate      280
Canonical height, for divisor of degree zero      285
Canonical height, is a quadratic form      218 454 461
Canonical height, lattice structure associated to      254
Canonical height, lower bound      287
Canonical height, on an elliptic surface      217 247 265 266 269 281 286
Canonical height, on Jacobian variety      271
Canonical height, over function fields is rational      219 247
Canonical height, parallelogram law      218
Canonical height, positive definite      218
Canonical height, regulator      273
Canonical height, standard properties      267
Castelnuovo’s criterion      236 344 352
Category of elliptic surfaces up to birational equivalence      206
Category of S-schemes      298
Category of varieties up to birational equivalence      205
centralizer      179
Character, Artin      405
Character, Swan      405
Characteristic polynomial of Probenius      172
Characteristic zero      187 189
Chinese remainder theorem      103 158
Circle group      129
Circle method      61
Class field theory      115 — 120; see also Complex multiplication
Class field theory of $\mathbb{Q}$       151 153
Class field theory, Artin map      117 118 154
Class field theory, idele group      119
Class field theory, idelic formulation      118 — 120
Class field theory, local      see Local class field theory
Class field theory, ray class field      117 118 120 129 135 180
Class field theory, reciprocity map      120 152 159 165 166 168 169 174
Class function      405
Class number      107
Class number, is finite      86
Class number, one      107 109 138 141
Class number, two      142 181
Closed fiber over a DVR      300
cm      see Complex multiplication
Codimension one, generic point of subscheme      328
Codimension one, regular in      311
Cohomology, -adic      172
Commutator subring of endomorphism ring      129
Complete intersection      328
Complete ring is Henselian      330
Complex Lie group      48
Complex multiplication      95
Complex multiplication, -torsion points are free module      102 109 138
Complex multiplication, action of ideal class group      99 122
Complex multiplication, additional references      95
Complex multiplication, associated Grssencharacter      165 168 174 175 184 185
Complex multiplication, by       96
Complex multiplication, by       96
Complex multiplication, by $\mathbb{Z}[(1 + \sqrt{— 7})/2]$       111 179
Complex multiplication, by $\mathbb{Z}[\rho$       102 107 177 180
Complex multiplication, by $\mathbb{Z}[\sqrt{— 2}$       110 180
Complex multiplication, by non-maximal order      160 180
Complex multiplication, computation of j      142 181
Complex multiplication, degree not square      143
Complex multiplication, degree of an isogeny      103 124
Complex multiplication, elliptic curve with CM by $R_{K}$       99
Complex multiplication, endomorphism ring      96 129
Complex multiplication, field of definition, of endomorphisms      106
Complex multiplication, field of definition,of isogenies      105
Complex multiplication, for abelian varieties      96 148
Complex multiplication, Galois action, on E      112 113 122 131

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