Arithmetic of algebraic curves:
Gespeichert in:
| Beteilige Person: | |
|---|---|
| Format: | Buch |
| Sprache: | Englisch Russisch |
| Veröffentlicht: |
New York, NY [u.a.]
Consultants Bureau
1994
|
| Schriftenreihe: | Monographs in contemporary mathematics
|
| Schlagwörter: | |
| Links: | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006857016&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| Beschreibung: | Aus dem Russ. übers. |
| Umfang: | XII, 422 S. |
| ISBN: | 0306110369 |
Internformat
MARC
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| 240 | 1 | 0 | |a Arifmetika algebraiceskich krivych |
| 245 | 1 | 0 | |a Arithmetic of algebraic curves |c Serguei A. Stepanov |
| 264 | 1 | |a New York, NY [u.a.] |b Consultants Bureau |c 1994 | |
| 300 | |a XII, 422 S. | ||
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| 490 | 0 | |a Monographs in contemporary mathematics | |
| 500 | |a Aus dem Russ. übers. | ||
| 650 | 7 | |a Algebraïsche krommen |2 gtt | |
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| 650 | 7 | |a Diofantische vergelijkingen |2 gtt | |
| 650 | 4 | |a Curves, Algebraic | |
| 650 | 4 | |a Curves, Elliptic | |
| 650 | 4 | |a Diophantine equations | |
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Datensatz im Suchindex
| _version_ | 1819250669538770944 |
|---|---|
| adam_text | Contents
Introduction 1
Chapter 1. Equations over Finite Fields 7
1.1. Congruences 7
1. Basic concepts 8
2. Congruences modulo a prime number 10
3. Algebraic congruences 11
Exercises 13
1.2. Congruences Relative to a Double Module and Finite Fields . . 21
l.The ring Fp[x] 21
2. The number of irreducible polynomials of degree n in FJ,[a;]. ¦ 23
3. Algebraic structure of finite fields 25
4. Automorphisms of the finite field Fq 27
5. The uniqueness of the field Fq 30
Exercises 31
1.3. Artin s Z Functions 35
1. Characters of finite Abelian groups 35
2. Characters of the field Fq 38
3. Artin s generating function 40
Exercises 46
1.4. Superelliptic Equations and the Artin Schreier Equation .... 50
1. Superelliptic equations and character sums 51
2. The number of .F, rational points on the curve f(x, y) = 0. . 53
3. Bounds for character sums with a polynomial 56
Exercises 59
ix
Chapter 2. Distribution of Quadratic Residues and
Nonresidues 71
2.1. The Results of Vinogradov and Burgess 71
1. The Vinogradov Polya theorem 71
2. Vinogradov s conjectures 73
3. Burgess theorem 75
Exercises 81
2.2. The Large Sieve and its Application to the Least Quadratic
Nonresidue Problem 88
1. The large sieve 88
2. Exceptional prime numbers 94
3. Linnik s theorem 95
Exercises 99
Historical Commentaries on Chapters 1 and 2 104
Chapter 3. Rational Points on Algebraic Curves 111
3.1. Rational Curves Ill
1. Plane algebraic curves Ill
2. Parametrization of curves 112
3. Algebraic curves of degree 2 114
4. Algebraic curves of degree n 3 119
Exercises 120
3.2. Elliptic Curves 125
1. Birational isomorphism of curves 125
2. Addition of points on elliptic curves 128
3. Mordell s theorem 130
4. The rank of an elliptic curve 135
Exercises 138
Chapter 4. The Riemann Roch Theorem 145
4.1. Affine and Projective Varieties 145
1. Affine algebraic sets 145
2. Regular maps 147
3. Rational functions on an algebraic variety 150
4. Projective and quasiprojective varieties 151
5. Nonsingular algebraic varieties 155
Exercises 159
4.2. Divisors of Algebraic Curves 162
1. The local ring of a point 162
2. Valuations 163
3. Divisors 171
Exercises 178
4.3. The Riemann Roch Theorem on an Algebraic Curve 182
1. The Riemann theorem 182
2. Repartitions 185
3. Differentials 189
4. The canonical class 194
Exercises 198
Chapter 5. The Riemann Hypothesis for Congruence
Zeta Functions 205
5.1. Zeta Functions of Algebraic Curves and Varieties 205
1. Rational points of a variety 205
2. Rational divisors on a curve 206
3. Zeta function of a curve 213
4. Zeta function of a variety 226
Exercises 233
5.2. The Number of Rational Points of an Algebraic Curve over a
Finite Field 240
1. Preliminary bound 241
2. The Hasse Weil bound 244
Exercises 245
Chapter 6. Integral Points on Curves and Nonstandard Arith¬
metic 251
6.1. Integral Points on Algebraic Curves 251
1. The Thue equation 251
2. Superelliptic equations 256
3. Integral points on curves of genus g 1 257
Exercises 259
6.2. Algebraic Systems and Models 268
1. Superstructures 271
2. Standard and nonstandard universes 275
3. Algebraic systems 277
4. The permanence principle 281
5. The orientation theorem 284
Exercises 294
6.3. Enlargements of Algebraic Number Fields 299
1. The arithmetic of an algebraic number field 299
2. The arithmetic of enlargement of an algebraic number field. . 300
3. Nonstandard prime divisors 302
4. Internal divisors 306
Exercises 314
Chapter 7. The Siegel Mahler Theorem 325
7.1. A Nonstandard Equivalent of the Siegel Mahler Theorem . . . 325
1. Functional divisors 325
2. Exceptional functional divisors 338
Exercises 346
7.2. Proof of the Siegel Mahler Theorem 360
1. The hyperelliptic case 360
2. The general case of curves of genus g 1 364
Exercises 370
Appendix. Hilbert s Tenth Problem 384
1. The informal computability 384
2. Turing s machines 385
3. Partial recursive functions 386
4. Church s thesis 387
5. Recursively enumerable sets 388
6. Diophantine sets and predicates 389
7. Positive aspects of the negative solution of Hilbert s 10th
Problem 389
References 391
Index 415
|
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| dewey-ones | 512 - Algebra |
| dewey-raw | 512/.72 |
| dewey-search | 512/.72 |
| dewey-sort | 3512 272 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV010305041 |
| illustrated | Not Illustrated |
| indexdate | 2024-12-20T09:51:42Z |
| institution | BVB |
| isbn | 0306110369 |
| language | English Russian |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-006857016 |
| oclc_num | 31655897 |
| open_access_boolean | |
| owner | DE-29T DE-355 DE-BY-UBR DE-20 DE-11 |
| owner_facet | DE-29T DE-355 DE-BY-UBR DE-20 DE-11 |
| physical | XII, 422 S. |
| publishDate | 1994 |
| publishDateSearch | 1994 |
| publishDateSort | 1994 |
| publisher | Consultants Bureau |
| record_format | marc |
| series2 | Monographs in contemporary mathematics |
| spellingShingle | Stepanov, Sergej A. Arithmetic of algebraic curves Algebraïsche krommen gtt Analyse diophantienne ram Courbes algébriques ram Courbes elliptiques ram Diofantische vergelijkingen gtt Curves, Algebraic Curves, Elliptic Diophantine equations Arithmetik (DE-588)4002919-0 gnd Diophantische Gleichung (DE-588)4012386-8 gnd Algebraische Kurve (DE-588)4001165-3 gnd |
| subject_GND | (DE-588)4002919-0 (DE-588)4012386-8 (DE-588)4001165-3 |
| title | Arithmetic of algebraic curves |
| title_alt | Arifmetika algebraiceskich krivych |
| title_auth | Arithmetic of algebraic curves |
| title_exact_search | Arithmetic of algebraic curves |
| title_full | Arithmetic of algebraic curves Serguei A. Stepanov |
| title_fullStr | Arithmetic of algebraic curves Serguei A. Stepanov |
| title_full_unstemmed | Arithmetic of algebraic curves Serguei A. Stepanov |
| title_short | Arithmetic of algebraic curves |
| title_sort | arithmetic of algebraic curves |
| topic | Algebraïsche krommen gtt Analyse diophantienne ram Courbes algébriques ram Courbes elliptiques ram Diofantische vergelijkingen gtt Curves, Algebraic Curves, Elliptic Diophantine equations Arithmetik (DE-588)4002919-0 gnd Diophantische Gleichung (DE-588)4012386-8 gnd Algebraische Kurve (DE-588)4001165-3 gnd |
| topic_facet | Algebraïsche krommen Analyse diophantienne Courbes algébriques Courbes elliptiques Diofantische vergelijkingen Curves, Algebraic Curves, Elliptic Diophantine equations Arithmetik Diophantische Gleichung Algebraische Kurve |
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