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

j-invariant, q-series      59 60 121 141 142 144 145 146 410
j-invariant, rationality properties      37 104
j-invariant, singular      104
j-invariant, supersingular      104
j-invariant, Tate curve $E_{q}$       423 425 438 473
j-invariant, transcendental values      108
j-invariant, valuation of      362 379
Jacobi sum      177
Jacobi, C.      62
Jacobian matrix      330 401
Jacobian variety      197 199 402
Jacobian variety, canonical height on      271
Jacobian variety, construction of      197
Jacobian variety, curve of genus two      287
Jacobian variety, dimension of      197
Jacobian variety, field of definition      199
Jacobian variety, is a functor      198
Jacobian variety, N $\acute{e}$ ron model      350
Jacobian variety, of a homogeneous space      199
Jacobian variety, of a hyperelliptic curve      199
Jacobian variety, of genus one curve      197
Jacobian variety, of genus zero curve      197
Jacobian variety, of hyperelliptic curve      486
Jacobi’s formula      60 62 65 90 409 468
Jordan normal form      179
Jugendtraum      96
Kernel of Artin map      117 118 179
Kodaira notation for special fiber      353
Kodaira reduction type      352 353 363 364
Kodaira, K.      287 352 486
Kodaira-Neron classification of fibers      350 352
Kouya, T.      272
Kramer, K.      385
Kraus, A.      486
Kronecker congruence relation      182
Kronecker Jugendtraum      96
Kronecker limit formula      484
Kronecker, L.      95
Kronecker-Weber theorem      95 129
Krull dimension      301 302
Kummer extension      383 446
Kummer sequence for an elliptic surface      193
Kummer sequence for fields      193
Kummer theory for function fields      191
Kuwata, M.      189 485
Kuyk, W.      389
l-adic cohomology      172
L-series      see also Dirichlet series
L-series, analytic continuation      83 93 94 173 176
L-series, attached to a cusp form      83 85 93
L-series, attached to a modular form      93
L-series, attached to a power series      80
L-series, attached to a twisted cusp form      94
L-series, elliptic curve      364
L-series, Euler product expansion      80 92 172 173
L-series, for j = 0 curve      178
L-series, functional equation      83 93 94 173 176 184
L-series, half-plane of convergence      83 172 183
L-series, Hecke      171 173 175 185
L-series, integral representation      84 85 94
L-series, leading coefficient      364
L-series, local of E at p      171 184 185
L-series, of an elliptic curve      171 172 175 183 185
L-series, relations satisfied by coefficients      80 92
L-series, residue of      93
L-series, special values of      93
Lang, S.      231 275 285 485
Lang’s conjecture, height lower bound      287
Laska, M.      364
Laska’s algorithm      364
Lattice function      74
Lattice function, associated to a modular function      74 79
Lattice, associated to a Weierstrass equation      35 88 89
Lattice, Euclidean      254
Lattice, Hecke correspondence on      68 90
Lattice, homothetic      6 9 14 37 101 103 161
Lattice, homothety correspondence on      68 90
Lattice, normalized      47
Lattice, normalized basis      7
Lattice, of an abelian variety      196 198
Lattice, of an elliptic curve      408
Lattice, oriented basis      6 9 71 72 73 89
Lattice, set of all ( $\mathcal{L}$ )      6 36 68 74
Lattice, sublattice      67
Lattice, sublattice of index n      71 72 73
Laurent series, for $\Gamma$       39 51
Laurent series, for $\zeta$       39
Laurent series, for F(u; q)      49 51
Laurent series, p-adic      429
Lefschetz principle      199 463
Legendre relation      41 465 469
Lehmer, D.H.      61
Lehner, J.      60
Lichtenbaum, S.      317 338 344
Lie group, complex      48
Lie group, real      420
Limit formula of Kronecker      484
Linear equivalence and height functions      256
Linear equivalence of divisors on a surface      232
Linear equivalence, intersection index      341
Linear fractional transformation      294
Linear group      292 396
Linearly equivalence      339
Lipman, J.      317
Littlewood, J.      61
Liu, Q.      390
Local class field theory      140 148 149 452
Local degree $n_{v}$       461
Local equation for a divisor      232 233
Local field, Artin character      405
Local field, different      386
Local field, higher ramification group      379
Local field, index function      405
Local field, inertia group      380
Local field, maximal unramified extension      381
Local field, Swan character      405
Local height function      429
Local height function, algorithm to compute      478 479
Local height function, associated to v      461
Local height function, comparison with naive height      478
Local height function, differential equation for      477
Local height function, duplication formula      456 475
Local height function, existence of      455 — 461 476
Local height function, existence of, archimedean absolute values      467
Local height function, existence of, non-archimedean absolute values      470 475
Local height function, integral over E( $\mathbb{C}$ )      477
Local height function, is a Green’s function      477
Local height function, is continuous      456
Local height function, is invariant under field extension      456 476
Local height function, is well-defined      456 470
Local height function, is zero for almost all v      461
Local height function, lower bound for      480
Local height function, multiplication formula      477
Local height function, on $E_{0}$ (K)      469 478
Local height function, on Tate curve $E_{q}$       473 475
Local height function, over $\mathbb{C}$       464 466 477 479 480
Local height function, over $\mathbb{R}$       479
Local height function, over non-archimedean fields      469 — 476
Local height function, over p-adic fields      469 — 476 478 479 480
Local height function, quasi-parallelogram law      467 476
Local height function, triplication formula      463
Local intersection index      233
Local intersection index, on $\mathbb{P}^{1}_{R}$       340 341
Local intersection index, on an arithmetic surface      339
Local L-series of E      171 184 185
Local N $\acute{e}$ ron height function      see Local height function
Local parameter      19 20 29 30 85
Local ring, is discrete valuation ring      232
Local ring, of a curve on an arithmetic surface      311 313 339
Local ring, of a surface at a curve      231

Local ring, of arithmetic surface      315
Local ring, of group scheme along identity      327
Local ring, regular      302 370 397
Local ring, regular of dimension two      315
localization      179
Locally compact field      476
Lockhart, P.      385
Logarithm, formal      150 182
Logarithm, principal branch      44 54 56 66
L’H $\hat{o}$ pital’s rule      45
magic      48
Main theorem of complex multiplication      157 159 166
Manin, Yu.      247 265 269 271 275 485
Mason, R.C.      277 278
Masser, D.      278
Matrix, incidence      350 402 403 486
Matrix, Jacobian      330 401
Maximal abelian extension, of exponent m unramified outside S      191
Maximal abelian extension, of quadratic imaginary field      135
Maximal unramified extension, abelian      see Hilbert class field
Maximal unramified extension, of a local field      381
Meromorphic function, j is a      23 35
Meromorphic function, on $\mathbb{P}^{1}$       35
Meromorphic function, on H      24
Meromorphic function, on X(1)      23 35
Mestre, J.-F.      272 486
Minimal discriminant      363 364 388 389
Minimal discriminant, greater than conductor      396
Minimal discriminant, of an elliptic surface      286
Minimal discriminant, of isogenous curves      453
Minimal discriminant, over 2-adic field      407
Minimal discriminant, over 3-adic field      406
Minimal discriminant, Szpiro’s conjecture      287 388
Minimal discriminant, valuation one      369 399
Minimal discriminant, valuation two      399
Minimal fibered surface      244
Minimal fibered surface, birational map is morphism      244
Minimal proper regular model, components of special fiber      389
Minimal proper regular model, connected component      326 361 401
Minimal proper regular model, identity component of special fiber      326 361
Minimal proper regular model, number of components of special fiber      364
Minimal proper regular model, of a curve      317 318 344
Minimal proper regular model, of an elliptic curve      325 332 335 337 350 361 400
Minimal proper regular model, reduction type      352 353 363 364
Minimal proper regular model, special fiber      350 352 403
Minimal surface, relatively      235
Minimal Weierstrass equation      321 403 478 480
Minimal Weierstrass equation, algorithm      364 403
Minimal Weierstrass equation, for p $\geq$ 3      406
Minimal Weierstrass equation, for p $\geq$ 5      405
Modular curve $X_{1}$ (11)      190 279
Modular curve X(1)      14
Modular curve X(1), affine part Y(1)      14
Modular curve X(1), complex structure on      16 20 26 28 35
Modular curve X(1), cusp      14
Modular curve X(1), differential forms on      23 28 87
Modular curve X(1), has genus zero      20 21
Modular curve X(1), is a Hausdorff space      18 19
Modular curve X(1), is a Riemann surface      20 32
Modular curve X(1), is compact      19
Modular curve X(1), is connected      21
Modular curve X(1), j-function on      23 34 36
Modular curve X(1), local parameter      20 21 29 30 85
Modular curve X(1), meromorphic functions on      23 35
Modular curve X(1), open cover of      20
Modular curve X(1), order of a differential form      28
Modular curve X(1), projection is open map      19
Modular curve X(1), topology on      16 18 19
Modular discriminant      see Discriminant
Modular form      25 915; cusp
Modular form, action of Hecke operator      76
Modular form, dimension of space of      31 87
Modular form, Eisenstein series is a      25 31 32
Modular form, Fourier coefficients of      82 92
Modular form, L-series attached to      93
Modular form, size of Fourier coefficients      82 92
Modular form, space of ( $M_{2k}$ )      31 87
Modular form, space of is generated by $G_{4}$ , $G_{6}$       88
Modular form, spaces with dimension one      33
Modular function      24 91; Modular Cusp
Modular function, action of Hecke operator      76 79 91
Modular function, associated differential form      28 31 87 91
Modular function, associated lattice function      74
Modular function, eigenfunction for Hecke operator      77
Modular function, formula for order      30
Modular function, Fourier series      24 76 145
Modular function, holomorphic at $\infty$       24
Modular function, j is a      34
Modular function, meromorphic at $\infty$       24
Modular function, of weight zero      34 35 182
Modular function, order at $\infty$       24
Modular function, value at $\infty$       24 32
Modular function, weight of      24
Modular function, why even weight      24
Modular group $\Gamma$ (1)      6 — 14; see also Action of $SL_{2}(\mathbb{Z})$ on H
Modular group $\Gamma$ (1), generated by S and T      14 24 62 85
Modular group $\Gamma$ (1), is a free product      14 85
Modular j-invariant      see j-invariant
Modular polynomial, $F_{n}$       144 146 181 182
Modular polynomial, $H_{n}$       144 146
Modular polynomial, $Ф_{n}$       181 182
Modular polynomial, examples      148
Modular polynomial, size of coefficients      181
Monster group      61
Montgomery, H.      487
Mordell conjecture      271
Mordell — Weil group, Euclidean structure      247
Mordell — Weil group, lattice structure      254
Mordell — Weil theorem, bound for rank      279
Mordell — Weil theorem, for abelian varieties      485
Mordell — Weil theorem, for elliptic surfaces      276
Mordell — Weil theorem, for function fields      230 279
Mordell — Weil theorem, for split elliptic surfaces      231 281
Mordell — Weil theorem, ineffective      277
Mordell — Weil theorem, over finitely generated fields      279
Mordell — Weil theorem, relative      231 281
Mordell — Weil theorem, weak      191 — 195 230
Mordell, L.J.      61 79
Morphism, $\acute{e}$ tale      306
Morphism, birational      235
Morphism, diagonal      300 309 327
Morphism, divisor associated to      258
Morphism, fiat      304 397
Morphism, graph of      300
Morphism, of S-schemes      297
Morphism, proper      305
Morphism, separated      305
Morphism, smooth      304 305 306 320
Morphism, universally closed      305
Multiple fibers      203
Multiplication by an idele      159 170
Multiplication map, continuous for v-adic topology      460
Multiplication map, on elliptic curve      310
Multiplication map, on group scheme      310 398
Multiplication-by-a; map on $\mathbb{Q}/\mathbb{Z}$       152 153
Multiplication-by-n map on an elliptic surface      266
Multiplicative group      128 151 398 399
Multiplicative group, exponential map      151
Multiplicative group, formal      149 431
Multiplicative group, over $\mathbb{Z}$       308
Multiplicative group, over a field      291 293 398
Multiplicative group, over R      308
Multiplicative group, over S      308
Multiplicative group, rational points of      292
Multiplicative group, S-valued points      397
Multiplicative group, Tate module of      382
Multiplicative group, torsion subgroup      151
Multiplicative reduction      287 369 388 399
Multiplicative reduction, conductor      381
Multiplicative reduction, N $\acute{e}$ ron model      403

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