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Husemoeller D. — Elliptic curves
Husemoeller D. — Elliptic curves



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Название: Elliptic curves

Автор: Husemoeller D.

Аннотация:

This book is an introduction to the theory of elliptic curves, ranging from its most elementary aspects to current research. The first part, which grew out of Tate's Haverford lectures, covers the elementary arithmetic theory of elliptic curves over the rationals. The next two chapters recast the arguments used in the proof of the Mordell theorem into the context of Galois cohomology and descent theory. This is followed by three chapters on the analytic theory of elliptic curves, including such topics as elliptic functions, theta functions, and modular functions. Next, the theory of endomorphisms and elliptic curves over infinite and local fields are discussed. The book then continues by providing a survey of results in the arithmetic theory, especially those related to the conjecture of the Birch and Swinnerton-Dyer. This new edition contains three new chapters which explore recent directions and extensions of the theory of elliptic curves and the addition of two new appendices. The first appendix, written by Stefan Theisan, examines the role of Calabi-Yau manifolds in string theory, while the second, by Otto Forster, discusses the use of elliptic curves in computing theory and coding theory. Dale Husemöller is a member of the faculty at the Max Planck Institute of Mathematics in Bonn.


Язык: en

Рубрика: Математика/Алгебра/Алгебраическая геометрия/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Издание: second edition

Год издания: 2004

Количество страниц: 487

Добавлена в каталог: 14.03.2005

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$d_{K/\mathbb{Q}}$ is the absolute discriminant      313
$E_{\infty}$-spectrum      443
$f_{E/K}$ is the conductor      313
$\Gamma$-function      315
$\varphi$-descent      166
$\zeta$-function      171
2-division point theta function      196
2-isogeny      95
A-valued G-cocycle      149
Abel — Jacobi theorem      205
Abelian varieties      240 345
Addition formulas for the Weierstrass p-function      177
Addition of two points      25
Additional structures on an elliptic curve      212
Additional structures on an elliptic curve, basis (P, Q) of $_N E$      212
Additional structures on an elliptic curve, cyclic subgroup C of order N      212
Additional structures on an elliptic curve, point P of order iV      212
Additive group      42
Admissible change of variables      68 71
Affine scheme      385
Affline modular curve      214
Algebraic group homomorphism      319
Algebraic Hecke character      319-321
Algebraic Hecke Grossencharakters      314
Algorithmic number theory      413
Almost complex structure      349 350
anomalies      407
Artin      255
Atiyah — Singer index formula      376
Automorphism of elliptic curves      34
Axioms for Chern classes      362
Bad reduction at p      109
Basic descent formalism      163
Basic theta function      193
Basic theta function of type $-z^{-1}$      205
Betti numbers      356
Bezout’s Theorem      50 52
Bianchi identity      364
Binomial series      180
Birch and Swinnerton-Dyer      326
Birch and Swinnerton-Dyer Conjecture over $\mathbb{Q}$      330
Birch conjecture      125
Bosons      405
Bott periodicity      442
Breuil      333
Calabi      346
Calabi — Yau manifold      346
Calabi — Yau manifolds      366
Calabi — Yau varieties      345
Canonical height      137 142
Canonical p-adic filtration of E      276
Carayol      339
Category      384 386
Category object      427
Category of category objects      429
Cauchy’s Theorem      169
Cech cocycle      365
CHARACTER      295
Characteristic polynomial of Frobenius      255
Characteristic polynomials of the l-adic representation $\rho$      299
Chern forms      364
Chord-tangent compositions      13
Chow’s Lemma      378\
Closed      352\
Closed string      404
Coates      326\
Coates and Wiles      326
Cobounding      150
Cocategory object      432
Coclosed      352
Cocycle formula      149
Cogroupoid      433
Common eigenform      337
Compact Kahler manifolds      352
Compactification      408
Compatible family      302
Complete intersection      342 371
Complete intersections in weighted projective spaces      370
Complex conjugation      351
Complex multiplication      233 242
Complex oriented      442
Complex structure      353
Complex torus      210 211
Complex torus, isogenous      210
Complex torus, isomorphic      210
Complex vectors fields      347
complexity      420
Composition law      66
Conductor      278 292 322 335
Conductor of an algebraic Hecke character      320
Conjecture of Calabi      346
Conjecture of Tate      305
Conjugation invariant polynomials      364
Connection      357 358
Connections      357
Conrad      333
Cotangent bundle      346
Criterion of Neron — Ogg — Safarevic      334
Cross section functor      385
Crude Hasse — Weil zeta function      310
Cryptography      417
Curvature      358 359
Curvature form      358
Cusp form      229 337
Cusp on a cubic curve      41
De Rham cohomology      351
Dedekind ring      291
Dedekind zeta function      316 326
Deformation data      339
Deformation theory      339
Deligne      255 312 323
Deligno–Serre      339
density      299
Derivative      331
Derivative of an L-function      331
Descent      157
Descent procedure      126
Deuring      269 309 322
Deuring’s theorem      321
diamond      333
Differential equation for $\wp(z)$      176
Diffie — Hellman key exchange      417
Diophantus of Alexandria      7
Dirichlet character      228 317
Dirichlet class formula      326
Dirichlet series      224 227 309 314
Dirichlet’s class number formula      328
Discrete logarithm      417
Discrete subgroup      167
Discriminant      70 71 119 178 193
Division point      233
Division polynomial      272
Divisor      173 371 388
Divisor class      389
Divisor class group      293
Double point on a cubic curve      41
Duality      410
Effective divisor      389 396
Eigenform      337
Einstein’s field equations      403
Eisenstein series      220 285
Elementary symmetric functions      363
Elkies      331
Elkies primes      423
Elliptic cohomology      425
Elliptic curve      1 14 20
Elliptic curve factorization      413
Elliptic curve, isogenous      271\
Elliptic curves      386
Elliptic curves over the real numbers      284
Elliptic curves, semistable      335
Elliptic curves, table of      289
Elliptic fibration      390 395
Elliptic function      169
Endomorphism      241
Endomorphism algebra      266
Endomorphism, characteristic polynomial of      241
Endomorphism, degree map of an      241
Endomorphism, trace of      241
Enriques classification      346
Enriques classification for surfaces      377
Euler number      218
Euler product      224
Euler product of degree n      225 226
Euler product, quadratic      226
Euler products      310
Euler systems of Kolyvagin      341
Exceptional curves      377
Exceptional points      392
Exponential sequence      373
Extension, normal      145
Exterior differential      351
Factorial ring      57
Faltings      9 304 306 334 335
Families of elliptic curves      383
Family of schemes      386
Fermat descent      127
Fermat equation      30
Fermat’s Last Theorem      333
fermions      405
Feynman diagrams      404
Fibration      390
Fibration of curves      390
Field $M_L$ of elliptic functions      172
Field, algebraically closed      145
First Birch and Swinnerton-Dyer cnjecture      326
Flach      341
Fniteness of the index      129
Formal group $\Phi(X, Y) of an elliptic curve E      251
Formal group law      250
Formal group of an elliptic curve      248
Formal logarithm      251
Fourier decomposition      229
Frey      331 333 335
Frey curve      336
Frobenius      421
Frobenius element      291
Frobenius elements      300
Frobenius endomorphism      254
Frobenius morphism      268
Frst Chern class      365 366
Frst Chern class of a line bundle      362
Frst Chern class of analytic/algebraic line bundles      373
Full subcategory of groupoids      430
Fulton      371
Function field      141
Functional equation      255
G-action      431
Galois action on homogeneous space      158
Galois cohomology      143 153
Galois cohomology classification of curves      155
Galois extension      145
Galois group      294
Galois representation      338
Galois stable lattices      304
Gamma factor      313
Gamma function      179
Gauss’s Lemma      59
Genus formula      390
Genus of $X_1(N)$ and $X_2(N)$      219
Genus of X(N)      219
Global field      141
Globally minimal normal cubic equation      293
Goldfeld      328
Good reduction      335 337
Good reduction at p      109
Gorenstein      341
Greenberg      327
Gross and Zagier      328
Grossencharacter of the type $(A_0)$      320
Grothendieck      255 363
Group object      430
Groupoid      429
Harmonic      352
Hasse      16 242 269
Hasse invariant      260 261
Hasse — Weil conjecture      309
Hasse — Weil L-function      309 312 313
Hasse — Weil zeta      311
Hecke      221 223 318
Hecke algebra      334 338
Hecke Grossencharacters      309
Hecke L-function      317
Hecke L-function with “Grossencharakter”      315
Hecke L-functions      309
Hecke operator      229
Hecke operators      336
Heegner point      328 331
Heegner points      328
Height h on $\mathbb{P}_m(k)$      135
Hermitian metric      353
Hermitian vector bundle      359
Hessian family      88 89
Hilbert’s “Theorem 90”      154
Hirzebruch      366
Hodge filtration      352
Hodge numbers      356
Hodge to de Rham spectral sequence      367
Holomorphic connection      358 360
Holomorphic elliptic function      169
Holomorphic manifold      350
Holomorphic stable vector bundle      408
Holonomy group      360
Homogeneous space      157
Homogeneous spaces over elliptic curves      157
Homology theory      441
Homothetic      168
Honda      272
Hopf algebroids      425
Hurwitz’s relation      218
Hypergeometric differential equation      182
Hypergeometric function      182 184
Hypergeometric series      181
Hyperplane      46
Ideal sheaves      387
Idele group      318
Imaginary quadratic field with class number      1 244
Indecomposable of canonical type      393
Index theorem for complex surfaces      376
Inertia subgroup      281
Intersection multiplicity      54
Intersection of the two quadratic surfaces      20—22 196
Intersection theory for plane curves      50
Invariant      70
Invariant differential      68 251
Irreducible      57
Irreducible plane algebraic curve      47
Isogeny      168 233 238 306
Isogeny between two complex tori      234
Isogeny, degree of      238
Isogeny, dual      236 239
Isogeny, index of an      210
Isomorphism      70 73 75 76
Isomorphism classes      386
J-function      178 215
j-values      333
j-values over $\mathbb{Q}$      333
Jacobi family      91 92
Jacobi q-parametrization      189
k-morphism of degree d      135
1 2
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