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Knapp A.W. — Elliptic Curves (MN-40)
Knapp A.W. — Elliptic Curves (MN-40)

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Название: Elliptic Curves (MN-40)

Автор: Knapp A.W.

Аннотация:

An elliptic curve is a particular kind of cubic equation in two variables whose projective solutions form a group. Modular forms are analytic functions in the upper half plane with certain transformation laws and growth properties. The two subjects — elliptic curves and modular forms — come together in Eichler-Shimura theory, which constructs elliptic curves out of modular forms of a special kind. The converse, that all rational elliptic curves arise this way, is called the Taniyama-Weil Conjecture and is known to imply Fermat's Last Theorem.

Elliptic curves and the modeular forms in the Eichler- Shimura theory both have associated L functions, and it is a consequence of the theory that the two kinds of L functions match. The theory covered by Anthony Knapp in this book is, therefore, a window into a broad expanse of mathematics — including class field theory, arithmetic algebraic geometry, and group representations — in which the concidence of L functions relates analysis and algebra in the most fundamental ways.

Developing, with many examples, the elementary theory of elliptic curves, the book goes on to the subject of modular forms and the first connections with elliptic curves. The last two chapters concern Eichler-Shimura theory, which establishes a much deeper relationship between the two subjects. No other book in print treats the basic theory of elliptic curves with only undergraduate mathematics, and no other explains Eichler-Shimura theory in such an accessible manner.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1992

Количество страниц: 448

Добавлена в каталог: 05.04.2008

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
Abelian, subvariety      368
Abelian, variety      367
Abel’s theorem      318
Addition      162 363
Addition, elliptic curve      11
Addition, formula      76
Adeles      407
Admissible change of variable      63 363
Affine algebraic set      342
Affine coordinate ring      342
Affine curve      25 343
Affine local coordinates      21 345
Affine problem      3
Affine variety      342
Algebraic integer      122 123
Algebraic numbers      122
Algebraic set, affine      342
Algebraic set, projective      344
Analytic continuation      166 167
Arithmetic-geometric mean      185 402
Associate Dirithlet characters      213
Associativity      12 67
Atkin — Lehner theorem      283 289
Automorphic function      333
Automorphic L function      xi 208
Bad prime      108
Bezout’s Theorem      27 47
Birational map      348
Birch and Swinnerton-Dyer conjecture      17 208 386
Canonical class      316 361
Canonical height      35 97
Canonical structure      333 341 349 357
Carayol      391
Change of variables, admissible      63 362
CHARACTER      200
Chord case      75
Chord-tangent composition rule      11 43
Class group, divisor      316 361
Class group, ideal      127
Class number      126
Complex multiplication      164 407
Complex torus      160 183 370
Conductor      213 390
Congruence subgroup      250 256
Congruent numbers      52 110 112 115
conic      25
Continuation, analytic      166 167
Convergent product      195
Coordinate ring      342
Coordinates, affine local      345
Correspondence      385
Cubic      25 40 56
Cubic, nonsingular      58
Cubic, singular      58
Curve, affine      25 343
Curve, elliptic      42
Curve, nonsingular      27
Curve, projective      345
Curve, projective plane      24
Curve, same      24
Curve, singular      27
Curve, smooth      27
Cusp      77 262
Cusp form      225 261
Defined at      344 346 348
Defined over      342 344 347 361 362 368
Degenerate      68
Degree      243 249 273 317 359 360
Descent      80
Differential holomorphic      313
Differential invariant      314
Differential meromorphic      312
DIMENSION      343 346
Dimension formula      235 272
Diophantus      3 6 7 10 50
Diophantus method      6 10 115 118
Dirichlel character      20
Dirichlel character, associate      213
Dirichlel character, conductor of      313
Dirichlel character, extension      113
Dirichlel character, primitive      213
Dirichlel character, principal      201
Dirichlel L function      xii 201
Dirichlet series      192
Dirichlet unit theorem      125
Dirichlet’s Theorem      xii 148 189
Discrete valuation ring      351
Discriminant      15 58 58 125 226
Divisor      316 360
Divisor class group      316 361
Divisor, principal      316 360
Dominant morphism      343
Dominate      392
Double series theorem      158
Doubling formula      76
Doubly penodir      1 52
Dual isogeny      309 365
Eichler — Shimura theory      xii 302 374 388 390 404
Eigenform      280
Elliptic curve      13 42 160 183 362
Elliptic curve, modular      221
Elliptic element      303
Elliptic function      152
Elliptic integral      174
Elliptic regulator      106
Equivalent eigenform      280
Equivalent ideals      126
Euler product      196 202 245 282 289 285
Euler product, irst degree      197
Euler product, second degree      198
Even total winding number      170
Extension of Dirichlet character      213
Fairly bad prime      108
Fermat      52 110 112
Fermat’s Last Theorem      xii 3 18 50 54 58 81 397 390
Fermat’s method of descent      80
Fermat’s problem for Mersenne      55 119
fibonacci      52
First degree Euler product      197
Flex      35
Fourier inversion formula      200 210
Fourier transform      200 310
Frey — Serre — Ribet      xii 18 399
Frobenius morphism      360 384
Frobenius morphism, $q^{th}$ power      360
Function element      166
Function field      312 319 342 346
Function field of dimension 1      349
Functional equation      209 216 240 270 289 386 387 388
Fundamental domain      228 260
Fundamental parallelogram      152
Gauss      18 5 402
Gauss sum      214 387
Genus      272 117 961 395
Global minimal      290
Good prime      108
Haase — Minkowski theorem      5
Hacke operator      244 275 320 321 323 372
Hasse principle      5 15 16
Hasse — Weil conjecture      18 387
Hasse — Weil L function      403
Hasse’s theorem      16 296
Hecke subgroup      256
Hecke — Petersson theorem      255 282
Hecke’s theorem      349 279
Height      95 97
Hessian matrix      30
Hilbert basis theorem      342
Hilbert nullstellensatz      342
Holomorphic at $\infty$      225
Holomorphic at the cusp      261 263
Holomorphic differential      312
Homogeneous      243 273
Homogeneous ideal      344
Homogeneous polynomial      22
Homology      317 321 392
Homomorphism      368
Ideal class group      127
Identity, elliptic curve      11
Igusa      374 390
Inflection point      32
Integral, elliptic      174
Intersection multiplicity      32
Invariant differential      314
inversion      269 285
Inversion problem      165
Irreducible      342 345
Isogenous      365 369
Isogeny      164 309 363 369
Isogeny, dual      309 369
Isomorphic elliptic curves      63
Isomorphic over      349
Isomorphic varieties      349
Isomorphism      368
J-invariant      65 226
Jacobi inversion theoren      319
Jacobian variety      310 318 361 369 370
L function      xi 17 201 238 267 295
L function, automorphic      xi 208
L function, Dirichlet      xii 201
L function, mulivic      xi 207
Langlands program      xiii
Lattice      343 318
Legendre symbol      373 298 393
Legendre’s theorem      5
Level      261 333 390
Lie group      376
Line      19 25
Line, at infinity      19
Line, same      19
Line, tangent      27
Linear system      316 361
Liouville theorem,      152 153
Local coordinate, affine      21 345
Local L factor      294
Local ring      343 346
Lutz — Nagell theorem      xi 15 130 144
Major’s theorem      15
Mellin transform      238
Meromorphic differential      312
Mestre — Oesterle      396
Method of descent      80
Method of Diophantus      6 10 115 119
Minimal Weierstraas equation      290
Minimal Weierstraas equation at prime p      290
Minimal Weierstraas equation, global      290
Minkowski      5 104
Model      938 341 349 357
Modular elliptic curve      221
Modular form      234 335 261
Modular form, unrestricted      224 261
Modular pair      273
Modular parametrization      310 390
Modular polynomial      335
Mordell — Weil theorem      xi 80 95 402
Mordell’s theorem      xi 14 80 95 402
Morphism      348 356
Morphism, dominant      349
Morphism, Frobenius      360 384
Motivic L funclion      xi
Multiplicative      196
Multiplicative, strictly      197
Multiplicity one theorem      283
Multiplicity, intersection      32
Nagell      xi 15 130 144
Naive height      95
Negative      12
Newform      283
Newton      10
Node      77
Noetherian      342
Nondegenerate      68
Nonsingular      25 27 58 161 343 347
Nonsplit case      79
Norm      123
Nullstellensatz      342
Number field      122
Odeal of variety      342 344
Oldform      283
Order      153
p-adic filtration      138
p-adic norm      134
p-integral      135
p-reduced      135
Parabolic element      303
Parallelogram, fundamental      152
Parallelogram, period      152
Period      151 184
Period lattice      153
Period parallelogram      152
Petersson inner product      242 252 280
Plane curve      24
Poincare      13 67
Point of inflection      35
Point, nonsingular      25 343 347
Point, singular      25 77
POINTS      19 25 342 344
Points at infinity      19
Poisson summation formula      211
Pole      300
Prescribed torsion      145
primitive      213
Principal axis theorem      37
Principal congruence subgroup      250 256
Principal Dirichlet character      201
Principal divisor      316 360
Product of ideals      126
Product of varieties      347
Product, convergent      195
Projective, algebraic art      344
Projective, closure      345
Projective, curve      345
Projective, n-space      344
Projective, plane      19
Projective, plane curve      24
Projective, problem      3
Projective, transformation      20 345
Projective, variety      345
Purely inseparable      358
Pythagorean triple      7
q-exparwton      225 263 265
Quadratic residue symbol      272 298 393
Ramified      359
Rank      15 102 107
Rational map      347
Rational points      19 25 342 344
Reciprocal roots      196
Reduction modulo p      130 134 364
Regular      344 346 348
Regulator      100
Remann surface      170 311 312 316
Representation theory      xiii 407
Result of rearranging      167
Resultant      44
Ribet      xii 18 396 399
Riemann xeta function      189 194
Riemann — Roch theorem      317 361
Same curve      24
Same line      19
Schwartz function      211
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