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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
Complex multiplication, Galois action, on j      112 122 166
Complex multiplication, generate ring of integers      95
Complex multiplication, good reduction      184
Complex multiplication, group of $\mathfrak{a}$-torsion points      102 135
Complex multiplication, isogenous curves      178 180
Complex multiplication, Iwasawa theory      96
Complex multiplication, j-invariant      121 122 166
Complex multiplication, j-invariant, in $\mathbb{Q}$      138
Complex multiplication, j-invariant, is algebraic      104
Complex multiplication, j-invariant, is integral      140 147 151 176 447
Complex multiplication, j-invariant, j = 0      102 107 177 180
Complex multiplication, j-invariant, j = 1728      101 107 138 185
Complex multiplication, j-invariant, j = 8000      110 180
Complex multiplication, j-invariant, j = —3375      111
Complex multiplication, L-series      171 175 185
Complex multiplication, lift of Frobenius map      130 132 162
Complex multiplication, main theorem      157 159 166
Complex multiplication, map F : Gal($\bar{K}$/K) $\rightarrow \mathcal{CL}(R_{K})$      112 121
Complex multiplication, number of points modulo $\beta$      175
Complex multiplication, of degree two      109 141 179
Complex multiplication, ordinary reduction      179
Complex multiplication, potential good reduction      140 447
Complex multiplication, reduction mod $\beta$      124 127 129 131
Complex multiplication, Tate module      179 186
Complex multiplication, torsion generates abelian extensions      108 135 148 149 166 168
Complex multiplication, transitive action, of Galois group      122
Complex multiplication, transitive action, of ideal class group      100 113
Complex multiplication, twist      183
Complex multiplication, Weber function      134 135
Complex multiplication, with $j\in \mathbb{R}$      142 179
Complex multiplication,by $\mathbb{Z}[i]$      101 107 109 138 185
Complex structure      19
Components of fiber, group of      363 364
Components of fiber, multiplicity of      313
Components of fiber, non-reduced      313
Components of fiber, number on special fiber      363 364 389
Components of fiber, of a divisor      231
Components of fiber, of multiplicity one      402
Components of fiber, reducible      313
Components of fiber, singular      313
Conductor, bounded      404
Conductor, constant under isogeny      404
Conductor, divisor of an elliptic surface      287
Conductor, exponent of      363 364 380 389
Conductor, exponent of, bound for      385
Conductor, exponent of, computation of      389 407
Conductor, exponent of, less than discriminant      396
Conductor, exponent of, maximum value      387 407
Conductor, exponent of, over 2-adic field      407
Conductor, exponent of, over 3-adic field      407
Conductor, independent of l      381 405
Conductor, of a Gr$\ddot{o}$ssencharacter      176
Conductor, of an abelian extension      117 118 123
Conductor, of an elliptic curve      379 — 388
Conductor, of an order      178
Conductor, of ray class field      117 118 129 135
Conductor, square-free      388
Conductor, Szpiro’s conjecture      287 388
Conductor, tame part      380 381 405
Conductor, wild part      380 381 405
Conductor-discriminant formula for ellip¬tic curves      389
Connected component of N$\acute{e}$ron model      326 361 401
Connected fibers      240 283 342 353
Contravariant functor      309
Correspondence      68
Correspondence, Hecke      68; see also Hecke operator
Correspondence, homothety      68
Correspondence, on set of lattices      68 90
Correspondence, ring of      70
Cox, D.      486
Cremona, J.E.      389
Criterion of N$\acute{e}$ron — Ogg — Shafarevich      362
Curve, ample divisor on      257 264
Curve, automorphism group      294
Curve, canonical divisor      27 87
Curve, differential form on      207
Curve, exceptional      236 344
Curve, height on      257 264 265
Curve, holomorphic differential on      198
Curve, homology of      198
Curve, hyperelliptic      287 486
Curve, Jacobian variety of      197 199 402
Curve, minimal proper regular model of      317 318 344 361
Curve, normal law on symmetric product      402
Curve, of genus two      287
Curve, Picard group of      192 195 199 402
Curve, projective      27 87
Curve, proper regular model of      317
Curve, symmetric product      229
Curve, uniformizer for      339
Curve;      see also Riemann surface
Cusp      287 370
Cusp form      25 91; modular
Cusp form, action of Hecke operator      76 78 79 92
Cusp form, basis of eigenfunctions      78 80 92
Cusp form, dimension of space of      31
Cusp form, discriminant is a      26 31
Cusp form, Fourier coefficients of      79 80 81
Cusp form, L-series attached to      83 85 93
Cusp form, simultaneous eigenfunction      79 80 92
Cusp form, size of Fourier coefficients      81
Cusp form, space of ($M^{0}_{2k}$)      31 78
Cusp form, spaces with dimension one      34
Cusp form, twisted by $\chi$      93
Cusp form, twisted L-series attached to      94
Cusp, of H*      14
Cusp, of X(1)      14
Cusp, stabilizer of      15
Cyclotomic extension, unramified      383
Cyclotomic field      95 128 151
Cyclotomic field, is abelian      108 128 151
Decomposition group      116 331
Dedekind $\eta$ function      65
Dedekind $\eta$ function, transformation formula for      67 90
Dedekind domain      113 179
Dedekind domain, arithmetic surface over      311
Dedekind domain, is regular scheme      302
Dedekind reciprocity      67 90
Dedekind sum s(x,y)      66 89
Dedekind, R.      67
Degree map on divisors      197
Degree one prime, in ideal class      118 123 161
Degree, non-square is CM      143
Degree, of a CM isogeny      103 124
Degree, of a divisor      232 282
Degree, of isogeny is preserved under reduc¬tion      124
Deligne, P.      61 81
Dem’janenko, V.A.      265 269 271 485
Descent theorem      230
Descent, faithfully flat      338
Determinant, Deuring, M.      95 175
Determinant, homomorphism of group varieties      292
Determinant, kernel of      292
Diagonal, height associated to      285
Diagonal, is complete intersection      328
Diagonal, morphism      300 309 327
Different      176 386
Differential form      207
Differential form, associated to a modular function      28 31 87 91
Differential form, divisor of      27
Differential form, holomorphic      27
Differential form, invariant      43 183
Differential form, of the first kind      43
Differential form, of the second kind      43
Differential form, of the third kind      43
Differential form, on a curve      198
Differential form, on a Riemann surface      27
Differential form, on elliptic curve      43
Differential form, on H      26
Differential form, on X(1)      23
Differential form, order of      27
Differential form, regular      27
Differential form, sheaf of relative      306
Dimension, of a point on a scheme      302
Dimension, of a scheme      302
Dimension, of Jacobian variety      197
Dimension, regular scheme of dimension one      302
Dirichlet character, primitive      93
Dirichlet series      see also L-series
Dirichlet series, analytic continuation      83 94
Dirichlet series, attached to a cusp form      83 85
Dirichlet series, attached to a power series      80
Dirichlet series, Euler product expansion      80 92 172 173
Dirichlet series, functional equation      83 94
Dirichlet series, half-plane of convergence      83 94
Dirichlet series, relations satisfied by coefficients      80 92
Dirichlet’s theorem on primes in progres¬sions      118 123
Discrete subgroup of $\mathbb{Q*}_p}$      422
Discrete subgroup of $\mathbb{Q}_p}$      422
Discrete valuation ring      328
Discrete valuation ring, arithmetic surface over      311
Discrete valuation ring, closed fiber over      300
Discrete valuation ring, generic fiber over      300
Discrete valuation ring, Henselian      330 401
Discrete valuation ring, Henselization of      331
Discrete valuation ring, smooth scheme over      304
Discrete valuation ring, special fiber over      300
Discrete valuation ring, strict Henselization of      331
Discrete valuation ring, strictly Henselian      330 402
Discrete valuation ring, unramified extension      399
Discriminant function, Fourier expansion      59 60 467 468
Discriminant, Fourier coefficients      482; see also Ramanujan $\tau$ function Ja¬cobi’s
Discriminant, is eigenfunction for Hecke operators      78 79
Discriminant, minimal      363 364 388 389 396
Discriminant, minimal, for elliptic surface      286
Discriminant, minimal, over 2-adic field      407
Discriminant, minimal, over 3-adic field      406
Discriminant, modular      26 31 32 62 89 135
Discriminant, of a cubic polynomial      62
Discriminant, of a Weierstrass equation      62 183 455 458 470 478
Discriminant, of isogenous curves      453
Discriminant, of Tate curve $E_{q}$      377 423 424 473
Discriminant, product expansion      60 62 90 409 468
Discriminant, vanishes only at $\infty$      32
Discriminant, weight 12 cusp form      26 31 32 34 66
div      see Divisor group
Division polynomial $\psi_{m}$      477
Divisor class group of $\mathbb{P}^{1}$      271
Divisor class group of a curve      see Picard group
Divisor class group of a function field      see Picard group
Divisor group      231
Divisor group, induced homomorphism      255
Divisor group, of a fibered surface      284
Divisor group, of a set      68
Divisor group, of a variety      255
Divisor group, of an arithmetic surface      339
Divisor, algebraic equivalence      254 285 486
Divisor, ample      233 257 258
Divisor, associated to a morphism      258
Divisor, canonical height      285
Divisor, components of      231
Divisor, degree of      232 282
Divisor, exceptional      344
Divisor, fibral      246 247 284 340
Divisor, group of      231
Divisor, height associated to      256 265
Divisor, horizontal      340
Divisor, integer part of      87
Divisor, intersection pairing      233 238
Divisor, is difference of very amples      261 284
Divisor, linear equivalence      232 339
Divisor, local equation      232 233
Divisor, minimal discriminant      286
Divisor, numerical equivalence      486
Divisor, of a differential form      27
Divisor, of a section      245
Divisor, of an elliptic function      45
Divisor, on a surface      231
Divisor, on an arithmetic surface      311 339
Divisor, positive      285 341
Divisor, principal      232 339
Divisor, self-intersection      234 238 243 283 342 349 351
Divisor, Serre’s theorem on ample      261
Divisor, transversal intersection      232 282
Divisor, very ample      257 261 284
Divisor, Weil      339
Domain of a rational map      204 279
Domain of morphism to group scheme      327 333 337
Dominant rational map      204 222 228
Dominant rational map, from smooth scheme to proper scheme      328
Dominant rational map, induces map on function fields      205
Dominant rational map, of elliptic surfaces      208
Dual graph of special fiber      353
Dual isogeny      67 125 229
Dual isogeny, of N$\acute{e}$ron model      400
Duplication formula      139 427 437
Duplication formula, for height      257
Duplication formula, for local height      456 471 475
Duplication formula, has degree four      458
Duplication formula, on an elliptic curve      210 214 457 471
Dynkin diagram      353
Effective bound for S-integer points      277 288
Eigenfunction, basis of $M^{0}_{2k}$      80 92
Eigenfunction, for Hecke operators      77
Eigenfunction, normalized      78 79 80 92 93
Eigenfunction, simultaneous for all $T_{2k}$’s      78 79 80 92 93
Eisenstein series      25 91
Eisenstein series, $E_{2k}$      58
Eisenstein series, $G_{2k}$      25 31 32 55
Eisenstein series, $g_{2}$ and $g_{3}$      26 35 58
Eisenstein series, $G_{4}$ and $G_{6}$ algebraically independent      88
Eisenstein series, $G_{4}$ and $G_{6}$ determine $\Lambda$      35 88
Eisenstein series, algebra generated by      88
Eisenstein series, Fourier coefficients of      58
Eisenstein series, is a modular form      25
Eisenstein series, is eigenfunction for Hecke operators      78 93
Eisenstein series, normalized      58
Eisenstein series, q-expansion of      55
Eisenstein series, special value of      178
Eisenstein series, value at $\infty$      25
Elkies, N.      487
Elliptic curve, automorphism group      183
Elliptic curve, bad reduction      185
Elliptic curve, classification of special fibers      350 352
Elliptic curve, conductor      363 364 388
Elliptic curve, Elliptic curve, addition formula      323
Elliptic curve, endomorphism of degree two      109 141
Elliptic curve, endomorphism ring R      98
Elliptic curve, endomorphisms are integral      448
Elliptic curve, existence of N$\acute{e}$ron model      325 332 335 337
Elliptic curve, field of definition      37 38
Elliptic curve, formal group      276
Elliptic curve, Galois conjugate      131
Elliptic curve, good reduction      184
Elliptic curve, Gr$\ddot{o}$ssencharacter of      165 168 174 175
Elliptic curve, group of $a$-torsion points      102 135
Elliptic curve, height over function field      212 213
Elliptic curve, homogeneous space of      199
Elliptic curve, invariant differential      43 97 134
Elliptic curve, is abelian variety      196
Elliptic curve, is group variety      291
Elliptic curve, isogeny      67 182 183
Elliptic curve, j- invariant, j = 8000      110 180
Elliptic curve, j- invariant, j = — 3375      111
Elliptic curve, j- invariant, non-integral has End(E) = $\mathbb{Z}$      447
Elliptic curve, j-invariant, j = 0      102 107 177 180 280
Elliptic curve, j-invariant, j = 1728      101 107 138 185 280
Elliptic curve, l-adic representation      405 445
Elliptic curve, L-series      171 172 175 183 185 364
Elliptic curve, map to $\mathbb{P}^{1}$      134
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