Elliptic curves:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York ; NY
Springer
[2004]
|
| Edition: | 2. edition |
| Series: | Graduate texts in mathematics
111 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis Klappentext |
| Item Description: | Includes bibliographical references (p. [465]-478) and index |
| Physical Description: | XXI, 487 Seiten Illustrationen, Diagramme |
| ISBN: | 0387954902 9780387954905 9780387215778 |
Staff View
MARC
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| 100 | 1 | |a Husemöller, Dale |0 (DE-588)117713058 |4 aut | |
| 245 | 1 | 0 | |a Elliptic curves |c Dale Husemöller: With appendices by Otto Forster, Ruth Lawrence, and Stefan Theisen |
| 250 | |a 2. edition | ||
| 264 | 1 | |a New York ; NY |b Springer |c [2004] | |
| 264 | 4 | |c © 2004 | |
| 300 | |a XXI, 487 Seiten |b Illustrationen, Diagramme | ||
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| 490 | 1 | |a Graduate texts in mathematics |v 111 | |
| 500 | |a Includes bibliographical references (p. [465]-478) and index | ||
| 650 | 4 | |a Courbes algébriques | |
| 650 | 4 | |a Courbes elliptiques | |
| 650 | 4 | |a Schémas en groupes - Mathématiques | |
| 650 | 4 | |a Curves, Algebraic | |
| 650 | 4 | |a Curves, Elliptic | |
| 650 | 4 | |a Group schemes (Mathematics) | |
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Record in the Search Index
| _version_ | 1804129305678053376 |
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| adam_text | Contents
Preface to the Second Edition vii
Preface to the First Edition ix
Acknowledgments to the Second Edition xi
Acknowledgments to the First Edition xiii
Introduction to Rational Points on Plane Curves 1
1 Rational Lines in the Projective Plane 2
2 Rational Points on Conies 4
3 Pythagoras, Diophantus, and Fermat 7
4 Rational Cubics and Mordell s Theorem 10
5 The Group Law on Cubic Curves and Elliptic Curves 13
6 Rational Points on Rational Curves. Faltings and the Mordell
Conjecture 17
7 Real and Complex Points on Elliptic Curves 19
8 The Elliptic Curve Group Law on the Intersection of Two Quadrics
in Projective Three Space 20
1 Elementary Properties of the Chord Tangent Group Law
on a Cubic Curve 23
1 Chord Tangent Computational Methods on a
Normal Cubic Curve 23
2 Illustrations of the Elliptic Curve Group Law 28
3 The Curves with Equations y2 = .v3 + ax and y2 = .v3 + a 34
4 Multiplication by 2 on an Elliptic Curve 38
5 Remarks on the Group Law on Singular Cubics 41
2 Plane Algebraic Curves 45
1 Projective Spaces 45
2 Irreducible Plane Algebraic Curves and Hypersurfaces 47
xvi Contents
3 Elements of Intersection Theory for Plane Curves 50
4 Multiple or Singular Points 52
Appendix to Chapter 2: Factorial Rings and Elimination Theory .... 57
1 Divisibility Properties of Factorial Rings 57
2 Factorial Properties of Polynomial Rings 59
3 Remarks on Valuations and Algebraic Curves 60
4 Resultant of Two Polynomials 61
3 Elliptic Curves and Their Isomorphisms 65
1 The Group Law on a Nonsingular Cubic 65
2 Normal Forms for Cubic Curves 67
3 The Discriminant and the Invariant y 70
4 Isomorphism Classification in Characteristics ^ 2, 3 73
5 Isomorphism Classification in Characteristic 3 75
6 Isomorphism Classification in Characteristic 2 76
7 Singular Cubic Curves 80
8 Parameterization of Curves in Characteristic Unequal to 2 or 3 82
4 Families of Elliptic Curves and Geometric Properties
of Torsion Points 85
1 The Legendre Family 85
2 Families of Curves with Points of Order 3: The Hessian Family .... 88
3 The Jacobi Family 91
4 Tate s Normal Form for a Cubic with a Torsion Point 92
5 An Explicit 2 Isogeny 95
6 Examples of Noncyclic Subgroups of Torsion Points 101
5 Reduction mod p and Torsion Points 103
1 Reduction mod p of Projective Space and Curves 103
2 Minimal Normal Forms for an Elliptic Curve 106
3 Good Reduction of Elliptic Curves 109
4 The Kernel of Reduction mod p and the p Adic Filtration Ill
5 Torsion in Elliptic Curves over Q: Nagell Lutz Theorem 115
6 Computability of Torsion Points on Elliptic Curves from Integrality
and Divisibility Properties of Coordinates 118
7 Bad Reduction and Potentially Good Reduction 120
8 Tate s Theorem on Good Reduction over the Rational Numbers .... 122
6 Proof of Mordell s Finite Generation Theorem 125
1 A Condition for Finite Generation of an Abelian Group 125
2 Fermat Descent and x4 + y4 = 1 127
3 Finiteness of (£(Q) : 2£(Q)) for E = E[a, b] 128
4 Finiteness of the Index (E(k) : 2E(k)) 129
5 Quasilinear and Quasiquadratic Maps 132
6 The General Notion of Height on Projective Space 135
Contents xvii
7 The Canonical Height and Norm on an Elliptic Curve 137
8 The Canonical Height on Projective Spaces over Global Fields .... 140
7 Galois Cohomology and Isomorphism Classification
of Elliptic Curves over Arbitrary Fields 143
1 Galois Theory: Theorems of Dedekind and Artin 143
2 Group Actions on Sets and Groups 146
3 Principal Homogeneous G Sets and the First Cohomology Set
Hl(G, A) 148
4 Long Exact Sequence in G Cohomology 151
5 Some Calculations with Galois Cohomology 153
6 Galois Cohomology Classification of Curves with Given j Invariant 155
8 Descent and Galois Cohomology 157
1 Homogeneous Spaces over Elliptic Curves 157
2 Primitive Descent Formalism 160
3 Basic Descent Formalism 163
9 Elliptic and Hypergeometric Functions 167
1 Quotients of the Complex Plane by Discrete Subgroups 167
2 Generalities on Elliptic Functions 169
3 The Weierstrass p Function 171
4 The Differential Equation for p(z) 174
5 Preliminaries on Hypergeometric Functions 179
6 Periods Associated with Elliptic Curves: Elliptic Integrals 183
10 Theta Functions 189
1 Jacobi g Parametrization: Application to Real Curves 189
2 Introduction to Theta Functions 193
3 Embeddings of a Torus by Theta Functions 195
4 Relation Between Theta Functions and Elliptic Functions 197
5 The Tate Curve 198
6 Introduction to Tate s Theory of /? Adic Theta Functions 203
11 Modular Functions 209
1 Isomorphism and Isogeny Classification of Complex Tori 209
2 Families of Elliptic Curves with Additional Structures 211
3 The Modular Curves X(N),X{(N), and X0(N) 215
4 Modular Functions 220
5 The L Function of a Modular Form 222
6 Elementary Properties of Euler Products 224
7 Modular Forms for ro(N),r (N), and F(N) 227
8 Hecke Operators: New Forms 229
9 Modular Polynomials and the Modular Equation 230
xviii Contents
12 Endomorphisms of Elliptic Curves 233
1 Isogenies and Division Points for Complex Tori 233
2 Symplectic Pairings on Lattices and Division Points 235
3 Isogenies in the General Case 237
4 Endomorphisms and Complex Multiplication 241
5 The Tate Module of an Elliptic Curve 245
6 Endomorphisms and the Tate Module 246
7 Expansions Near the Origin and the Formal Group 248
13 Elliptic Curves over Finite Fields 253
1 The Riemann Hypothesis for Elliptic Curves over a Finite Field .... 253
2 Generalities on Zeta Functions of Curves over a Finite Field 256
3 Definition of Supersingular Elliptic Curves 259
4 Number of Supersingular Elliptic Curves 263
5 Points of Order p and Supersingular Curves 265
6 The Endomorphism Algebra and Supersingular Curves 266
7 Summary of Criteria for a Curve To Be Supersingular 268
8 Tate s Description of Homomorphisms 270
9 Division Polynomial 272
14 Elliptic Curves over Local Fields 275
1 The Canonical p Adic Filtration on the Points of an Elliptic Curve
over a Local Field 275
2 The Neron Minimal Model 277
3 Galois Criterion of Good Reduction of Neron Ogg Safarevic 280
4 Elliptic Curves over the Real Numbers 284
15 Elliptic Curves over Global Fields and £ Adic Representations 291
1 Minimal Discriminant Normal Cubic Forms
over a Dedekind Ring 291
2 Generalities on ^ Adic Representations 293
3 Galois Representations and the Neron Ogg Safarevic Criterion in
the Global Case 296
4 Ramification Properties of £ Adic Representations of Number
Fields: Cebotarev s Density Theorem 298
5 Rationality Properties of Frobenius Elements in £ Adic
Representations: Variation of t 301
6 Weight Properties of Frobenius Elements in £ Adic
Representations: Faltings Finiteness Theorem 303
7 Tate s Conjecture, Safarevic s Theorem, and Faltings Proof 305
8 Image of € Adic Representations of Elliptic Curves: Serre s Open
Image Theorem 307
Contents xix
16 L Function of an Elliptic Curve and Its Analytic Continuation 309
1 Remarks on Analytic Methods in Arithmetic 309
2 Zeta Functions of Curves over Q 310
3 Hasse Weil L Function and the Functional Equation 312
4 Classical Abelian L Functions and Their Functional Equations 315
5 Grossencharacters and Hecke L Functions 318
6 Deuring s Theorem on the L Function of an Elliptic Curve with
Complex Multiplication 321
7 Eichler Shimura Theory 322
8 The Modular Curve Conjecture 324
17 Remarks on the Birch and Swinnerton Dyer Conjecture 325
1 The Conjecture Relating Rank and Order of Zero 325
2 Rank Conjecture for Curves with Complex Multiplication I, by
Coates and Wiles 326
3 Rank Conjecture for Curves with Complex Multiplication II, by
Greenberg and Rohrlich 327
4 Rank Conjecture for Modular Curves by Gross and Zagier 328
5 Goldfeld s Work on the Class Number Problem and Its Relation to
the Birch and Swinnerton Dyer Conjecture 328
6 The Conjecture of Birch and Swinnerton Dyer on the Leading Term 329
7 Heegner Points and the Derivative of the L function at s = 1, after
Gross and Zagier 330
8 Remarks On Postscript: October 1986 331
18 Remarks on the Modular Elliptic Curves Conjecture and
Fermat s Last Theorem 333
1 Semistable Curves and Tate Modules 334
2 The Frey Curve and the Reduction of Fermat Equation to Modular
Elliptic Curves over Q 335
3 Modular Elliptic Curves and the Hecke Algebra 336
4 Hecke Algebras and Tate Modules of Modular Elliptic Curves 338
5 Special Properties of mod 3 Representations 339
6 Deformation Theory and £ Adic Representations 339
7 Properties of the Universal Deformation Ring 341
8 Remarks on the Proof of the Opposite Inequality 342
9 Survey of the Nonsemistable Case of the Modular Curve Conjecture 342
19 Higher Dimensional Analogs of Elliptic Curves:
Calabi Yau Varieties 345
1 Smooth Manifolds: Real Differential Geometry 347
2 Complex Analytic Manifolds: Complex Differential Geometry 349
3 Kahler Manifolds 352
4 Connections, Curvature, and Holonomy 356
5 Projective Spaces, Characteristic Classes, and Curvature 361
xx Contents
6 Characterizations of Calabi Yau Manifolds: First Examples 366
7 Examples of Calabi Yau Varieties from Toric Geometry 369
8 Line Bundles and Divisors: Picard and Neron Severi Groups 371
9 Numerical Invariants of Si faces 374
10 Enriques Classification for Surfaces 377
11 Introduction to K3 Surfaces 378
20 Families of Elliptic Curves 383
1 Algebraic and Analytic Geometry 384
2 Morphisms Into Projective Spaces Determined by Line Bundles,
Divisors, and Linear Systems 387
3 Fibrations Especially Surfaces Over Curves 390
4 Generalities on Elliptic Fibrations of Surfaces Over Curves 392
5 Elliptic K3 Surfaces 395
6 Fibrations of 3 Dimensional Calabi Yau Varieties 397
7 Three Examples of Three Dimensional Calabi Yau Hypersurfaces
in Weight Projective Four Space and Their Fibrings 400
Appendix I: Calabi Yau Manifolds and String Theory 403
Stefan Theisen
Why String Theory? 403
Basic Properties 404
String Theories in Ten Dimensions 406
Compactification 407
Duality 409
Summary 411
Appendix II: Elliptic Curves in Algorithmic Number Theory and
Cryptography 413
Otto Forster
1 Applications in Algorithmic Number Theory 413
1.1 Factorization 413
1.2 Deterministic Primality Tests 415
2 Elliptic Curves in Cryptography 417
2.1 The Discrete Logarithm 417
2.2 Diffie Hellman Key Exchange 417
2.3 Digital Signatures 418
2.4 Algorithms for the Discrete Logarithm 419
2.5 Counting the Number of Points 421
2.6 Schoof s Algorithm 421
2.7 Elkies Primes 423
References 424
Contents xxi
Appendix III: Elliptic Curves and Topological Modular Forms 425
1 Categories in a Category 427
2 Groupoids in a Category 429
3 Cocategories over Commutative Algebras: Hopf Algebroids 431
4 The Category WT(R) and the Weierstrass Hopf Algebroid 434
5 Morphisms of Hopf Algebroids: Modular Forms 438
6 The Role of the Formal Group in the Relation Between Elliptic
Curves and General Cohomology Theory 441
7 The Cohomology Theory or Spectrum tmf 443
References 444
Appendix IV: Guide to the Exercises 445
Ruth Lawrence
References 465
List of Notation 479
Index 481
Dale
Husemöller
Elliptic Curves
Second Edition
This book is an introduction to the theory of elliptic curves, ranging from
elementary topics to current research. The first chapters, which grew out of
Tate s
Haverford
Lectures, cover the arithmetic theory of elliptic curves over
the field of rational numbers.This theory is then recast into the powerful and
more general language of Galois cohomology and descent theory. An analytic
section of the book includes such topics as elliptic functions, theta functions,
and modular functions. Next, the book discusses the theory of elliptic curves
over finite and local fields and provides a survey of results in the global arith¬
metic theory, especially those related to the conjecture of Birch and
Swinnerton-Dyer.
This new edition contains three new chapters. The first is an outline of Wiles s
proof of Fermat s Last Theorem.The two additional chapters concern higher-
dimensional analogues of elliptic curves, including
КЗ
surfaces and Calabi-Yau
manifolds.Three new appendices explore recent applications of elliptic curves
and their generalizations. The first, written by Stefan Theisen, examines the
role of Calabi-Yau manifolds and elliptic curves in string theory, while the sec¬
ond, by
Otto Forster,
discusses the use of elliptic curves in computing theory
and coding theory. The third appendix explains the role of elliptic curves in
homotopy theory.
About the First Edition:
All in all the book is well written, and can serve as basis for a student seminar
on the subject.
—
G. Faltings,
Zentralblatt
|
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| classification_rvk | SK 240 |
| ctrlnum | (OCoLC)49672279 (DE-599)BVBBV014486527 |
| dewey-full | 516.352 516.3/52 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 516 - Geometry |
| dewey-raw | 516.352 516.3/52 |
| dewey-search | 516.352 516.3/52 |
| dewey-sort | 3516.352 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | 2. edition |
| format | Book |
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| id | DE-604.BV014486527 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T19:03:02Z |
| institution | BVB |
| isbn | 0387954902 9780387954905 9780387215778 |
| language | English |
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| oclc_num | 49672279 |
| open_access_boolean | |
| owner | DE-703 DE-19 DE-BY-UBM DE-634 DE-20 DE-355 DE-BY-UBR DE-11 DE-188 DE-83 |
| owner_facet | DE-703 DE-19 DE-BY-UBM DE-634 DE-20 DE-355 DE-BY-UBR DE-11 DE-188 DE-83 |
| physical | XXI, 487 Seiten Illustrationen, Diagramme |
| publishDate | 2004 |
| publishDateSearch | 2004 |
| publishDateSort | 2004 |
| publisher | Springer |
| record_format | marc |
| series | Graduate texts in mathematics |
| series2 | Graduate texts in mathematics |
| spelling | Husemöller, Dale (DE-588)117713058 aut Elliptic curves Dale Husemöller: With appendices by Otto Forster, Ruth Lawrence, and Stefan Theisen 2. edition New York ; NY Springer [2004] © 2004 XXI, 487 Seiten Illustrationen, Diagramme txt rdacontent n rdamedia nc rdacarrier Graduate texts in mathematics 111 Includes bibliographical references (p. [465]-478) and index Courbes algébriques Courbes elliptiques Schémas en groupes - Mathématiques Curves, Algebraic Curves, Elliptic Group schemes (Mathematics) Elliptische Kurve (DE-588)4014487-2 gnd rswk-swf Algebraische Geometrie (DE-588)4001161-6 gnd rswk-swf Elliptische Kurve (DE-588)4014487-2 s Algebraische Geometrie (DE-588)4001161-6 s 1\p DE-604 Erscheint auch als Online-Ausgabe 978-0-387-21577-8 Graduate texts in mathematics 111 (DE-604)BV000000067 111 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009879716&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009879716&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Husemöller, Dale Elliptic curves Graduate texts in mathematics Courbes algébriques Courbes elliptiques Schémas en groupes - Mathématiques Curves, Algebraic Curves, Elliptic Group schemes (Mathematics) Elliptische Kurve (DE-588)4014487-2 gnd Algebraische Geometrie (DE-588)4001161-6 gnd |
| subject_GND | (DE-588)4014487-2 (DE-588)4001161-6 |
| title | Elliptic curves |
| title_auth | Elliptic curves |
| title_exact_search | Elliptic curves |
| title_full | Elliptic curves Dale Husemöller: With appendices by Otto Forster, Ruth Lawrence, and Stefan Theisen |
| title_fullStr | Elliptic curves Dale Husemöller: With appendices by Otto Forster, Ruth Lawrence, and Stefan Theisen |
| title_full_unstemmed | Elliptic curves Dale Husemöller: With appendices by Otto Forster, Ruth Lawrence, and Stefan Theisen |
| title_short | Elliptic curves |
| title_sort | elliptic curves |
| topic | Courbes algébriques Courbes elliptiques Schémas en groupes - Mathématiques Curves, Algebraic Curves, Elliptic Group schemes (Mathematics) Elliptische Kurve (DE-588)4014487-2 gnd Algebraische Geometrie (DE-588)4001161-6 gnd |
| topic_facet | Courbes algébriques Courbes elliptiques Schémas en groupes - Mathématiques Curves, Algebraic Curves, Elliptic Group schemes (Mathematics) Elliptische Kurve Algebraische Geometrie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009879716&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009879716&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000000067 |
| work_keys_str_mv | AT husemollerdale ellipticcurves |