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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
Generic fiber, of an arithmetic surface      311
Generic fiber, of an elliptic surface      206
Generic fiber, over a DVR      300
Generic point      304 328
Generic point, of a scheme      300
Generic point, of Spec(R)      300
Genus, arithmetic      342 351
Genus, in flat family      343
Genus, of a smooth curve      27 87
Genus, of the modular curve X(1)      20 21
Genus, two, special fiber of curve of      355
Genus, zero      343 352
Global sections      397
Good fiber      203
Good reduction      184 287 321 329 383 388
Good reduction, conductor      381
Good reduction, Gr$\ddot{o}$ssencharacter is unramified      168
Good reduction, potential      140 148 151 176 383
Good reduction, torsion for      453
Gr$\ddot{o}$ssencharacter      156 165 168 173 184
Gr$\ddot{o}$ssencharacter, conductor of      176
Gr$\ddot{o}$ssencharacter, for j = 0 curve      177
Gr$\ddot{o}$ssencharacter, L-series      171 173 175 185
Gr$\ddot{o}$ssencharacter, of twisted curve      183
Gr$\ddot{o}$ssencharacter, on $\mathbb{Q}$      156
Gr$\ddot{o}$ssencharacter, reduction modulo $\beta$      174
Gr$\ddot{o}$ssencharacter, unramified      168 173 174
Graded algebra      27 88
Graph of a morphism      300
Green’s function      477
Group chunk      198 334
Group cohomology      193
Group cohomology, of a cyclic group      421 449
Group law induced by normal law      334 337
Group law is normal law      334
Group of components, of a group variety      292
Group of components, of N$\acute{e}$ron model      350 362 363 364 379 402 486
Group of components, of special fiber      363 364
Group of sections to an elliptic surface      202 203 210 298
Group scheme      306 398
Group scheme, action      321 326 400
Group scheme, additive group      307 397 398 399
Group scheme, associated functor      309
Group scheme, associated to normal law      334 337
Group scheme, associativity      307
Group scheme, diagonal morphism      309
Group scheme, extension of group law      332 335
Group scheme, finite      397
Group scheme, identity component of special fiber      326 361
Group scheme, identity section      306 326
Group scheme, in characteristic p      397
Group scheme, inverse map      306
Group scheme, local ring along identity      327
Group scheme, morphisms to      327 333 337
Group scheme, multiplication map      310 398
Group scheme, multiplicative group      308 397 398 399
Group scheme, N$\acute{e}$ron model      319
Group scheme, of roots of unity      397 398
Group scheme, proper      327
Group scheme, set of T-valued points      309 397
Group scheme, smooth part of Weierstrass equation      321 362 378 399 400
Group scheme, smooth Weierstrass equation      329
Group scheme, translation map      310 336 337 400
Group variety      291
Group variety, abelian      196
Group variety, additive group      291 293 398
Group variety, algebraic      115
Group variety, defined over K      291
Group variety, determinant homomorphism      292
Group variety, elliptic curve      291 293 398
Group variety, general linear group      291
Group variety, homomorphism      292
Group variety, identity component      115 292
Group variety, irreducible components      292
Group variety, is a group scheme      307
Group variety, is non-singular      292
Group variety, Jacobian      402
Group variety, linear group      292 396
Group variety, multiplicative group      291 293 398
Group variety, of dimension one      293 398
Group variety, orthogonal group      396
Group variety, rational points of      292
Group variety, special fiber of N$\acute{e}$eron model      361
Group variety, special linear group      292
Group variety, special orthogonal group      396
Group variety, symplectic group      396
Group, algebraic      see Group variety
Group, pro-p      150 182
Group, sporadic      61
Group, topological      85
Half-plane of convergence, of L-series attached to cusp form      83
Half-plane of convergence, of L-series attached to elliptic curve      183
Hardy, G.H.      61
Hasse, H.      95
Hausdorff space      16 18 19 85
Hecke algebra      70
Hecke algebra, is commutative      71
Hecke algebra, of $\Gamma$(1)      70
Hecke L-series      173
Hecke operator      67 — 74; see also Hecke algebra
Hecke operator, action on a cusp form      76 78 79 92
Hecke operator, action on a lattice function      74
Hecke operator, action on a modular form      76
Hecke operator, action on a modular function      76 79 91
Hecke operator, action on Fourier coefficients      76 78 79
Hecke operator, as shifting operator      76
Hecke operator, commutativity of      71
Hecke operator, definition of      68
Hecke operator, discriminant is eigenfunction      78
Hecke operator, eigenfunction for      77
Hecke operator, explicit formula for      71 73 74 90
Hecke operator, recursive formula for      90
Hecke operator, relations satisfied by      68 79 90
Hecke operator, self-adjoint      92
Hecke operator, simultaneous eigenfunction for all      78 80 92 93
Hecke operator, T(p)      74 77
Hecke, E.      81 173 484
Heegner      141
Height Machine      256
Height Machine, standard properties      267 268
Height Machine, verification of      262
Height, additivity      256 261
Height, and linear equivalence      256
Height, associated to a divisor      256 265
Height, associated to a morphism      258
Height, associated to a positive divisor      285
Height, associated to an ample divisor      257
Height, associated to diagonal of curves      285
Height, associated to exceptional curve      285
Height, canonical      217 247 265 266 269 281 286
Height, duplication formula      213 257 280
Height, explicit O(1) estimate      280
Height, finitely many points of bounded      257 265
Height, for algebraically equivalent divisors      285
Height, functoriality      256
Height, geometric transformation properties      213
Height, infinitely many elements of bounded      213 220
Height, N$\acute{e}$ron — Tate      217 247 265 266 269 281 286;
Height, normalization      256
Height, on a function field      212
Height, on a variety      256
Height, on an elliptic surface      212 213 265
Height, on curves      257 264 265
Height, on projective space      255
Height, parallelogram law      213
Height, quasi-parallelogram law      280
Height, regulator      273
Height, specialization of      265 266 269 281
Height, sum of local      212
Henselian ring      330 401
Henselian ring, complete ring is      330
Henselian ring, strictly      330 402
Henselian ring, surjectivity of reduction      330 333 337 401
Henselization, fraction field      332
Henselization, of a DVR      331
Henselization, strict      337
Henselization, universal mapping property      332 401
Hensel’s Lemma      330 331 401 443
Hensel’s lemma, multi-variable version      401
Hermitian inner product      92
Higher ramification group      379
Higher ramification group, index function for      405
Higher ramification group, inertia group      380
Higher ramification group, is normal      380
Higher ramification group, over 2-adic field      404
Higher ramification group, over 3-adic field      404
Hilbert class field      95 118 130 132 142 181
Hilbert class field, abelian extension of      134
Hilbert class field, Artin map for      118
Hilbert class field, Galois group of      118
Hilbert class field, generated by j(E)      121 122 166
Hilbert class field, of $\mathbb{Q}(\sqrt{ — 15})$      180
Hilbert class field, of $\mathbb{Q}(\sqrt{ — 23})$      180
Hilbert class field, of quadratic imaginary field      121 122
Hilbert irreducibility theorem      272
Hilbert theorem      90 193 421
Hindry, M.      277 278 486 487
Holomorphic differential form      27 207
Holomorphic differential form, on a curve      198
Homogeneous space of an elliptic curve      199
Homology group      37 43
Homology group, of a curve      198
Homomorphism of group varieties      292
Homomorphism of groups      427
Homothetic lattices      6 9 14 37 101 103 161
Homothety operator      68 90
Homothety operator, relations satisfied by      68
Horizontal curve      237
Horizontal curve, intersection with      283
Horizontal divisor      237
Horizontal divisor, on an arithmetic surface      340
Hurwitz, A.      101 294 485
Hyperelliptic curve, integer points on      277
Hyperelliptic curve, Jacobian variety      199 287 486
Hyperelliptic curve, of genus two      287
Hyperelliptic curve, over function field      277
Hyperelliptic curve, Picard group      287
Ideal class group      99 180;
Ideal class group, acts on $\mathcal{ELL}(R_{K})$      99
Ideal class group, acts on $\mathcal{ELL}(R_{K})$ transitively      100 113
Ideal class group, algorithm to compute      85
Ideal class group, class contains degree one primes      118 123
Ideal class group, is finite      86
Ideal class group, is Picard group      192
Ideal class group, of a function field      192
Ideal class group, of quadratic imaginary field      85
Ideal of an idele      119 152 159
Ideal, principal      132
Idele group      119
Idele group, characterization of ray class fields      120 162
Idele group, class field theory using      118 — 120
Idele group, contains $K*_{v}$      119
Idele group, contains K*      119
Idele group, norm map      119
Idele group, topology on      119
Idele, ideal of      119 152 159
Idele, multiplication by      159 170
Identity component of a group variety      115 292
Identity component of N$\acute{e}$ron model is Weierstrass equation      362 378
Identity element N$\acute{e}$ron model      326
Identity section of a group scheme      306
Image of a rational map      204 279
Incidence matrix      240 241 243 283 350 402 403 486
Incomplete gamma function      93
Index function for higher ramification groups      405
Inert prime      184
Inertia group      120 149 169 331 380 445
Inertia group, absolute      380
Inner product, Hermitian      92
Inner product, Petersson      92
Inner product, positive definite      92
Inseparable isogeny      127
Integer points, effective methods      277 288
Integer points, on an elliptic surface      275
Integers of a function field      275
Integral scheme      311
Integral, elliptic      178
Integral, not path independent      198
Intersection index, Arakelov      344
Intersection index, computation of      283
Intersection index, example      349
Intersection index, linear equivalence      341
Intersection index, local      233 283
Intersection index, on $\mathbb{P}^{1}_{R}$      340 341
Intersection index, on an arithmetic surface      339 341
Intersection index, sum of local indices      234
Intersection index, symmetric      341
Intersection pairing      233
Intersection pairing, Arakelov      344
Intersection pairing, incidence matrix      240 241 243 283
Intersection pairing, on an arithmetic surface      341 342 353
Intersection pairing, on fibered surface      238
Intersection pairing, symmetric      341
Intersection, self      234 238 243 283 342 349 351
Intersection, transversal      232 282
Invariant differential      43 97 134 183
Inverse on a group scheme      306
Irreducibility theorem of Hilbert      272
Irreducible topological space      280
Isogeny      67 182
Isogeny, between CM curves      178 180
Isogeny, between Tate curves      453
Isogeny, comparison of discriminants      453
Isogeny, conductors are equal      404
Isogeny, degree preserved under reduction      124
Isogeny, dual      67 125 229 400
Isogeny, equations for      183
Isogeny, field of definition of      105
Isogeny, is a homomorphism      396
Isogeny, lift of Frobenius map      130 132 162
Isogeny, of N$\acute{e}$ron model      400
Isogeny, purely inseparable      127
Isogeny, reduction mod $\beta$      124 127 129
Isomorphism birational      204
Iwasawa module      186
Iwasawa theory      96
j-invariant, classifies elliptic curves      36
j-invariant, CM in $\mathbb{Q}$      107 138
j-invariant, computation of      142 181
j-invariant, divisibility properties of Fourier coefficients      60
j-invariant, elliptic surface with constant      286
j-invariant, Galois conjugate of      112
j-invariant, generates Hilbert class field      121 122 166
j-invariant, growth properties of Fourier coefficients      61
j-invariant, in $\mathbb{Z}$      141
j-invariant, integral      383 405
j-invariant, integrality properties      140 151 447
j-invariant, j = 0      102 107 180 280
j-invariant, j = 1728      101 107 138 280
j-invariant, j = 1728, Gr$\ddot{o}$ssencharacter      185
j-invariant, j = 1728, number of points modulo p      185
j-invariant, j = 8000      110 180
j-invariant, j = — 3375      111
j-invariant, map X(1)$\rightarrow\mathbb{P}^{1}$(C)      23 34 36 88
j-invariant, modular      34 88 121 143
j-invariant, modular function of weight zero      34 112 144
j-invariant, non-integral      382 405
j-invariant, of an elliptic surface      280 288
j-invariant, of CM curve is algebraic      104
j-invariant, of CM curve is integral      140 147 151 447
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