Variations on a theme of Euler: quadratic forms, elliptic curves, and Hopf maps
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English Japanese |
| Veröffentlicht: |
New York [u.a.]
Plenum Press
1994
|
| Schriftenreihe: | The university series in mathematics
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Aus dem Japan. übers. |
| Beschreibung: | XI, 347 S. graph. Darst. |
| ISBN: | 0306447894 |
Internformat
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| 240 | 1 | 0 | |a Euler no shudai ni yoru hensōkyoku |
| 245 | 1 | 0 | |a Variations on a theme of Euler |b quadratic forms, elliptic curves, and Hopf maps |c Takashi Ono |
| 264 | 1 | |a New York [u.a.] |b Plenum Press |c 1994 | |
| 300 | |a XI, 347 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a The university series in mathematics | |
| 500 | |a Aus dem Japan. übers. | ||
| 650 | 7 | |a Elliptische oppervlakken |2 gtt | |
| 650 | 7 | |a Kwadratische vormen |2 gtt | |
| 650 | 4 | |a Curves, Elliptic | |
| 650 | 4 | |a Forms, Quadratic | |
| 650 | 4 | |a Number theory | |
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Datensatz im Suchindex
| _version_ | 1804124606342103040 |
|---|---|
| adam_text | Contents
I. Introduction 1
1.1. X2+Y2 = Z2 1
1.1.1. Elementary Method 1
1.1.2. Use of Gaussian Field 4
1.1.3. Use of Hilbert Theorem 90 4
1.1.4. Use of Trigonometric Functions 5
1.1.5. Geometric Method 6
1.1.6. Review of Section 1.1 7
1.2. X2+Y2=U2,X2 Y2=V2 8
1.2.1. Fibonacci Fermat Theorem 8
1.2.2. Fermat s Proof 10
1.2.3. Application of the Theorem 12
1.2.4. Summary 13
References 14
1. Quadratic Forms 15
1.1. Basic Facts 15
1.1.1. Bilinear Maps and Quadratic Maps 15
1.1.2. Groups Acting on Q(X, Y) 17
1.1.3. Case Y=K 18
1.2. Orthogonality 21
1.2.1. Rank and Orthogonal Basis 21
1.2.2. Isomorphism and Direct Sum of Quadratic Spaces 25
1.3. Witt Theorems 27
1.3.1. Hyperbolic Plane 27
1.3.2. Witt Theorems 29
1.3.3. Case K=R 34
vii
viii Contents
1.3.4. Cayley s Parametrization 35
References 38
2. Algebraic Varieties 39
2.1. Affine Algebraic Varieties 39
2.1.1. Algebraic Sets and Ideals of Polynomials .... 39
2.1.2. Noetherian Spaces 42
2.1.3. Coordinate Rings, Rational Functions, and
Local Rings 44
2.1.4. Polynomial Maps 45
2.1.5. Regular Maps and Rational Maps 46
2.2. Affine Varieties Defined by Quadratic Forms 50
2.2.1. Irreducibility of Quadratic Hypersurfaces .... 50
2.2.2. Rationality of Quadratic Hypersurfaces .... 52
2.2.3. Irreducibility and Rationality of Rotation Groups 56
2.3. Projective Algebraic Varieties 59
2.3.1. Projective Algebraic Sets 59
2.3.2. Coordinate Rings, Rational Functions, and
Local Rings 63
2.3.3. Affine Varieties and Projective Varieties .... 64
2.3.4. Regular Maps and Rational Maps 67
2.4. Projective Varieties Defined by Quadratic Forms ... 70
2.4.1. Projective Quadratic Hypersurfaces 70
2.4.2. Separable Pairs of Quadratic Forms 71
2.4.3. Separable Pencils of Quadratic Forms 74
2.4.4. Intersection of Two Quadratic Hypersurfaces... 75
2.4.5. E(M,N) 77
2.4.6. C(M,N) 80
References 82
3. Plane Algebraic Curves 83
3.1. Affine Plane Curves 83
3.1.1. Simple Points, Singular Points, and Tangent Lines 83
3.1.2. Multiplicities and Local Rings 85
3.1.3. Intersection Numbers 89
3.2. Projective Plane Curves 93
3.2.1. Basic Facts 93
3.2.2. Bezout s Theorem 94
3.2.3. Noether s Theorem 97
3.2.4. Multiple Points 101
3.3. Plane Cubics 104
3.3.1. Flexes 104
Contents ix
3.3.2. Normal Forms and Invariants of Nonsingular Cubics 107
3.3.3. Group Structure of a Nonsingular Cubic . . . . Ill
3.3.4. Invariant and Group Structure of C(M, N) ... 114
References 121
4. Space Elliptic Curves 123
4.1. Theta Functions 123
4.1.1. Introduction of Theta Functions 123
4.1.2. Definition of Theta Functions 125
4.1.3. Zeros of Theta Functions 127
4.1.4. Product Expansions of Theta Functions .... 129
4.1.5. 9t = 9l + 9$ 134
4.1.6. Addition Formulas for Theta Functions .... 136
4.2. Theta Functions and E(M, N) 141
4.2.1. A Map 0 141
4.2.2. Group Structure of £ ( 1, k2) 147
4.2.3. Group Structure of E(M, N) 151
4.3. E(M,N)Q 153
4.3.1. Mordell Weil Theorem 153
4.3.2. EtOT(M,N)Q 154
4.3.3. To Find k Such That p(*r) 0 157
4.3.4. To Find k Such That p(xr) = 0 159
References 164
5. Quadratic Spherical Maps 165
5.1. Definitions and Examples 165
5.1.1. Definitions 165
5.1.2. Low Dimensional Cases 166
5.2. HopfMaps 171
5.2.1. Definition of a Hopf Map 171
5.2.2. Examples of Hopf Maps 173
5.3. Hopf Maps in Euclidean Spaces 178
5.3.1. Eigenvalues of Quadratic Spherical Maps and
HopfMaps 178
5.3.2. Applications of Eigenvalues 185
5.3.3. Wood Theorem 186
5.3.4. Hopf Map of the First Kind 190
5.3.5. Some Examples 192
References 198
6. Hurwitz Problem 199
6.1. Transformation of the Problem 199
x Contents
6.1.1. Basic Definitions 199
6.1.2. Algebras 200
6.1.3. Clifford Algebras 201
6.1.4. Transformation of the Hurwitz Problem .... 202
6.2. Structure of Algebras 206
6.2.1. Simple Algebras 206
6.2.2. Tensor Product of Algebras 210
6.2.3. Definition of F Algebra and Examples .... 217
6.2.4. Tensor Product of Z2 Algebras 221
6.3. Solution of the Problem (Case of Euclidean Spaces) . . 224
6.3.1. Structure of Clifford Algebras 224
6.3.2. Solution of Hurwitz Problem for Euclidean Spaces 233
6.3.3. Partial Solutions of the Hurwitz Problem in the
General Case 244
6.3.4. Some Examples 253
References 256
7. Arithmetic of Quadratic Maps 257
7.1. Hopf Fibration S3 S2 over Z 258
7.1.1. Sum of Two Squares 258
7.1.2. Contents 260
7.1.3. Gaussian Field and Hopf Fibration S3 +S2 over Z 264
7.2. Hopf Fibration S7 54 over Z 272
7.2.1. Announcement of Results 272
7.2.2. Arithmetic of Quaternions 273
7.2.3. The Number of Quaternions with a Given Norm 279
7.2.4. Proof of a Theorem on the Hopf Fibration S7 »S4 285
7.3. Hopf Fibration S15 S8 over Z 291
7.3.1. Hurwitz Triple and the Map/ 291
7.3.2. The Map /and the Map h 293
7.3.3. The Family Z£ 295
7.3.4. Hopf Fibration SI5 S8 over Z 300
References 303
Answers and Hints to Selected Exercises 305
Appendix 1. Euler s Elements of Algebra 321
A 1.1. HowDidEuler Write Elements of Algebra 321
A1.2. Contents of Elements of Algebra 321
A1.3. On E(M,N) 331
A 1.4. Chronological Table and Record 334
References 337
Contents xi
Appendix 2. A Short Survey of Subsequent Research on Congruent
Numbers 339
Masanari Kida
References 344
Index 345
|
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| illustrated | Illustrated |
| indexdate | 2024-07-09T17:48:20Z |
| institution | BVB |
| isbn | 0306447894 |
| language | English Japanese |
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| physical | XI, 347 S. graph. Darst. |
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| publishDateSearch | 1994 |
| publishDateSort | 1994 |
| publisher | Plenum Press |
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| series2 | The university series in mathematics |
| spelling | Ono, Takashi Verfasser aut Euler no shudai ni yoru hensōkyoku Variations on a theme of Euler quadratic forms, elliptic curves, and Hopf maps Takashi Ono New York [u.a.] Plenum Press 1994 XI, 347 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier The university series in mathematics Aus dem Japan. übers. Elliptische oppervlakken gtt Kwadratische vormen gtt Curves, Elliptic Forms, Quadratic Number theory Elliptische Kurve (DE-588)4014487-2 gnd rswk-swf Quadratische Form (DE-588)4128297-8 gnd rswk-swf Zahlentheorie (DE-588)4067277-3 gnd rswk-swf Quadratische Form (DE-588)4128297-8 s Zahlentheorie (DE-588)4067277-3 s DE-604 Elliptische Kurve (DE-588)4014487-2 s HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006778621&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Ono, Takashi Variations on a theme of Euler quadratic forms, elliptic curves, and Hopf maps Elliptische oppervlakken gtt Kwadratische vormen gtt Curves, Elliptic Forms, Quadratic Number theory Elliptische Kurve (DE-588)4014487-2 gnd Quadratische Form (DE-588)4128297-8 gnd Zahlentheorie (DE-588)4067277-3 gnd |
| subject_GND | (DE-588)4014487-2 (DE-588)4128297-8 (DE-588)4067277-3 |
| title | Variations on a theme of Euler quadratic forms, elliptic curves, and Hopf maps |
| title_alt | Euler no shudai ni yoru hensōkyoku |
| title_auth | Variations on a theme of Euler quadratic forms, elliptic curves, and Hopf maps |
| title_exact_search | Variations on a theme of Euler quadratic forms, elliptic curves, and Hopf maps |
| title_full | Variations on a theme of Euler quadratic forms, elliptic curves, and Hopf maps Takashi Ono |
| title_fullStr | Variations on a theme of Euler quadratic forms, elliptic curves, and Hopf maps Takashi Ono |
| title_full_unstemmed | Variations on a theme of Euler quadratic forms, elliptic curves, and Hopf maps Takashi Ono |
| title_short | Variations on a theme of Euler |
| title_sort | variations on a theme of euler quadratic forms elliptic curves and hopf maps |
| title_sub | quadratic forms, elliptic curves, and Hopf maps |
| topic | Elliptische oppervlakken gtt Kwadratische vormen gtt Curves, Elliptic Forms, Quadratic Number theory Elliptische Kurve (DE-588)4014487-2 gnd Quadratische Form (DE-588)4128297-8 gnd Zahlentheorie (DE-588)4067277-3 gnd |
| topic_facet | Elliptische oppervlakken Kwadratische vormen Curves, Elliptic Forms, Quadratic Number theory Elliptische Kurve Quadratische Form Zahlentheorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006778621&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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