Topics in the theory of algebraic function fields:
Presents an examination that explains both the similarities and fundamental differences between function fields and number fields, including many exercises and examples. This book serves as a text for a graduate course in number theory or an advanced graduate topics course.
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English Spanish |
| Veröffentlicht: |
Boston ; Basel ; Berlin
Birkhäuser
2006
|
| Schriftenreihe: | Mathematics: theory & applications
|
| Schlagworte: | |
| Online-Zugang: | Inhaltstext Inhaltsverzeichnis |
| Zusammenfassung: | Presents an examination that explains both the similarities and fundamental differences between function fields and number fields, including many exercises and examples. This book serves as a text for a graduate course in number theory or an advanced graduate topics course. |
| Beschreibung: | Literaturverz. S. 439 - 646 |
| Beschreibung: | XVI, 652 S. 24 cm |
| ISBN: | 9780817644802 0817644806 |
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| 240 | 1 | 0 | |a Introducción a la teoría de las funciones algebraicas |
| 245 | 1 | 0 | |a Topics in the theory of algebraic function fields |c Gabriel Daniel Villa Salvador |
| 264 | 1 | |a Boston ; Basel ; Berlin |b Birkhäuser |c 2006 | |
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| 650 | 4 | |a Algebraic functions | |
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Datensatz im Suchindex
| _version_ | 1804136416432619520 |
|---|---|
| adam_text | Contents
Preface vii
1 Algebraic and Numerical Antecedents 1
1.1 Algebraic and Transcendental Extensions 1
1.2 Absolute Values over Q 3
1.3 Riemann Surfaces 8
1.4 Exercises 11
2 Algebraic Function Fields of One Variable 13
2.1 The Field of Constants 14
2.2 Valuations, Places, and Valuation Rings 16
2.3 Absolute Values and Completions 26
2.4 Valuations in Rational Function Fields 36
2.5 Artin s Approximation Theorem 43
2.6 Exercises 52
3 The Riemann Roch Theorem 55
3.1 Divisors 55
3.2 Principal Divisors and Class Groups 61
3.3 Repartitions or Adeles 67
3.4 Differentials 72
3.5 The Riemann Roch Theorem and Its Applications 81
3.6 Exercises 88
4 Examples 93
4.1 Fields of Rational Functions and Function Fields of Genus 0 93
4.2 Elliptic Function Fields and Function Fields of Genus 1 101
4.3 Quadratic Extensions of k(x) and Computation of the Genus 105
4.4 Exercises 1ll
xiv Contents
5 Extensions and Galois Theory 113
5.1 Extensions of Function Fields 113
5.2 Galois Extensions of Function Fields 118
5.3 Divisors in an Extension 128
5.4 Completions and Galois Theory 132
5.5 Integral Bases 138
5.6 Different and Discriminant 147
5.7 Dedekind Domains 150
5.7.1 Different and Discriminant in Dedekind Domains 154
5.7.2 Discrete Valuation Rings and Computation of the Different ... 158
5.8 Ramification in Artin Schreier and Kummer Extensions 164
5.9 Ramification Groups 180
5.10 Exercises 186
6 Congruence Function Fields 191
6.1 Constant Extensions 191
6.2 Prime Divisors in Constant Extensions 193
6.3 Zeta Functions and L Series 195
6.4 Functional Equations 200
6.5 Exercises 207
7 The Riemann Hypothesis 209
7.1 The Number of Prime Divisors of Degree 1 209
7.2 Proof of the Riemann hypothesis 215
7.3 Consequences of the Riemann Hypothesis 222
7.4 Function Fields with Small Class Number 227
7.5 The Class Numbers of Congruence Function Fields 231
7.6 The Analogue of the Brauer Siegel Theorem 234
7.7 Exercises 237
8 Constant and Separable Extensions 239
8.1 Linearly Disjoint Extensions 239
8.2 Separable and Separably Generated Extensions 244
8.3 Regular Extensions 250
8.4 Constant Extensions 253
8.5 Genus Change in Constant Extensions 265
8.6 Inseparable Function Fields 276
8.7 Exercises 281
9 The Riemann Hurwitz Formula 283
9.1 The Differential dx in k(x) 283
9.2 Trace and Cotrace of Differentials 289
9.3 Hasse Differentials and Residues 292
9.4 The Genus Formula 307
9.5 Genus Change in Inseparable Extensions 311
Contents xv
9.6 Examples 325
9.6.1 Function Fields of Genus 0 325
9.6.2 Function Fields of Genus 1 330
9.6.3 The Automorphism Group of an Elliptic Function Field 337
9.6.4 Hyperelliptic Function Fields 344
9.7 Exercises 351
10 Cryptography and Function Fields 353
10.1 Introduction 353
10.2 Symmetric and Asymmetric Cryptosystems 354
10.3 Finite Field Cryptosystems 356
10.3.1 The Discrete Logarithm Problem 357
10.3.2 The Diffie Hellman Key Exchange Method and the Digital
Signature Algorithm (DSA) 357
10.4 Elliptic Function Fields Cryptosystems 358
10.4.1 Key Exchange Elliptic Cryptosystems 359
10.5 The ElGamal Cryptosystem 360
10.5.1 Digital Signatures 361
10.6 Hyperelliptic Cryptosystems 363
10.7 Reduced Divisors over Finite Fields 367
10.8 Implementation of Hyperelliptic Cryptosystems 370
10.9 Exercises 374
11 Introduction to Class Field Theory 377
11.1 Introduction 377
11.2 Cebotarev s Density Theorem 378
11.3 Inverse Limits and Profinite Groups 388
11.4 Infinite Galois Theory 400
11.5 Results on Global Class Field Theory 409
11.6 Results on Local Class Field Theory 411
11.7 Exercises 411
12 Cyclotomic Function Fields 415
12.1 Introduction 415
12.2 Basic Facts 416
12.3 Cyclotomic Function Fields 422
12.4 Arithmetic of Cyclotomic Function Fields 429
12.4.1 Newton Polygons 430
12.4.2 Abhyankar s Lemma 433
12.4.3 Ramification at p^ 435
12.5 The Artin Symbol in Cyclotomic Function Fields 438
12.6 Dirichlet Characters 448
12.7 Different and Genus 461
12.8 The Maximal Abelian Extension of K 463
12.8.1 E/K 463
xvi Contents
12.8.2 KT/K 464
12.8.3 Loc/K 469
12.8.4 A = EKtLoo 470
12.9 The Analogue of the Brauer Siegel Theorem 478
12.10 Exercises 480
13 Drinfeld Modules 487
13.1 Introduction 487
13.2 Additive Polynomials and the Carlitz Module 488
13.3 Characteristic, Rank, and Height of Drinfeld Modules 490
13.4 Existence of Drinfeld Modules. Lattices 496
13.5 Explicit Class Field Theory 504
13.5.1 Class Number One Case 505
13.5.2 General Class Number Case 507
13.5.3 The Narrow Class Field //+ 512
13.5.4 The Hilbert Class Field HA 516
13.5.5 Explicit Class Fields and Ray Class Fields 518
13.6 Drinfeld Modules and Cryptography 521
13.6.1 Drinfeld Module Version of the Diffie Hellman Cryptosystem 522
13.6.2 The Gillard et al. Drinfeld Cryptosystem 522
13.7 Exercises 523
14 Automorphisms and Galois Theory 527
14.1 The Castelnuovo Severi Inequality 527
14.2 Weierstrass Points 532
14.2.1 Hasse Schmidt Differentials 534
14.2.2 The Wronskian 542
14.2.3 Arithmetic Theory of Weierstrass Points 551
14.2.4 Gap Sequences of Hyperelliptic Function Fields 561
14.2.5 Fields with Nonclassical Gap Sequence 566
14.3 Automorphism Groups of Algebraic Function Fields 570
14.4 Properties of Automorphisms of Function Fields 583
14.5 Exercises 593
A Cohomology of Groups 597
A. 1 Definitions and Basic Results 597
A.2 Homology and Cohomology in Low Dimensions 615
A.3 Tate Cohomology Groups 624
A.4 Cohomology of Cyclic Groups 627
A.5 Exercises 631
Notations 635
References 639
Index 647
|
| adam_txt |
Contents
Preface vii
1 Algebraic and Numerical Antecedents 1
1.1 Algebraic and Transcendental Extensions 1
1.2 Absolute Values over Q 3
1.3 Riemann Surfaces 8
1.4 Exercises 11
2 Algebraic Function Fields of One Variable 13
2.1 The Field of Constants 14
2.2 Valuations, Places, and Valuation Rings 16
2.3 Absolute Values and Completions 26
2.4 Valuations in Rational Function Fields 36
2.5 Artin's Approximation Theorem 43
2.6 Exercises 52
3 The Riemann Roch Theorem 55
3.1 Divisors 55
3.2 Principal Divisors and Class Groups 61
3.3 Repartitions or Adeles 67
3.4 Differentials 72
3.5 The Riemann Roch Theorem and Its Applications 81
3.6 Exercises 88
4 Examples 93
4.1 Fields of Rational Functions and Function Fields of Genus 0 93
4.2 Elliptic Function Fields and Function Fields of Genus 1 101
4.3 Quadratic Extensions of k(x) and Computation of the Genus 105
4.4 Exercises 1ll
xiv Contents
5 Extensions and Galois Theory 113
5.1 Extensions of Function Fields 113
5.2 Galois Extensions of Function Fields 118
5.3 Divisors in an Extension 128
5.4 Completions and Galois Theory 132
5.5 Integral Bases 138
5.6 Different and Discriminant 147
5.7 Dedekind Domains 150
5.7.1 Different and Discriminant in Dedekind Domains 154
5.7.2 Discrete Valuation Rings and Computation of the Different . 158
5.8 Ramification in Artin Schreier and Kummer Extensions 164
5.9 Ramification Groups 180
5.10 Exercises 186
6 Congruence Function Fields 191
6.1 Constant Extensions 191
6.2 Prime Divisors in Constant Extensions 193
6.3 Zeta Functions and L Series 195
6.4 Functional Equations 200
6.5 Exercises 207
7 The Riemann Hypothesis 209
7.1 The Number of Prime Divisors of Degree 1 209
7.2 Proof of the Riemann hypothesis 215
7.3 Consequences of the Riemann Hypothesis 222
7.4 Function Fields with Small Class Number 227
7.5 The Class Numbers of Congruence Function Fields 231
7.6 The Analogue of the Brauer Siegel Theorem 234
7.7 Exercises 237
8 Constant and Separable Extensions 239
8.1 Linearly Disjoint Extensions 239
8.2 Separable and Separably Generated Extensions 244
8.3 Regular Extensions 250
8.4 Constant Extensions 253
8.5 Genus Change in Constant Extensions 265
8.6 Inseparable Function Fields 276
8.7 Exercises 281
9 The Riemann Hurwitz Formula 283
9.1 The Differential dx in k(x) 283
9.2 Trace and Cotrace of Differentials 289
9.3 Hasse Differentials and Residues 292
9.4 The Genus Formula 307
9.5 Genus Change in Inseparable Extensions 311
Contents xv
9.6 Examples 325
9.6.1 Function Fields of Genus 0 325
9.6.2 Function Fields of Genus 1 330
9.6.3 The Automorphism Group of an Elliptic Function Field 337
9.6.4 Hyperelliptic Function Fields 344
9.7 Exercises 351
10 Cryptography and Function Fields 353
10.1 Introduction 353
10.2 Symmetric and Asymmetric Cryptosystems 354
10.3 Finite Field Cryptosystems 356
10.3.1 The Discrete Logarithm Problem 357
10.3.2 The Diffie Hellman Key Exchange Method and the Digital
Signature Algorithm (DSA) 357
10.4 Elliptic Function Fields Cryptosystems 358
10.4.1 Key Exchange Elliptic Cryptosystems 359
10.5 The ElGamal Cryptosystem 360
10.5.1 Digital Signatures 361
10.6 Hyperelliptic Cryptosystems 363
10.7 Reduced Divisors over Finite Fields 367
10.8 Implementation of Hyperelliptic Cryptosystems 370
10.9 Exercises 374
11 Introduction to Class Field Theory 377
11.1 Introduction 377
11.2 Cebotarev's Density Theorem 378
11.3 Inverse Limits and Profinite Groups 388
11.4 Infinite Galois Theory 400
11.5 Results on Global Class Field Theory 409
11.6 Results on Local Class Field Theory 411
11.7 Exercises 411
12 Cyclotomic Function Fields 415
12.1 Introduction 415
12.2 Basic Facts 416
12.3 Cyclotomic Function Fields 422
12.4 Arithmetic of Cyclotomic Function Fields 429
12.4.1 Newton Polygons 430
12.4.2 Abhyankar's Lemma 433
12.4.3 Ramification at p^ 435
12.5 The Artin Symbol in Cyclotomic Function Fields 438
12.6 Dirichlet Characters 448
12.7 Different and Genus 461
12.8 The Maximal Abelian Extension of K 463
12.8.1 E/K 463
xvi Contents
12.8.2 KT/K 464
12.8.3 Loc/K 469
12.8.4 A = EKtLoo 470
12.9 The Analogue of the Brauer Siegel Theorem 478
12.10 Exercises 480
13 Drinfeld Modules 487
13.1 Introduction 487
13.2 Additive Polynomials and the Carlitz Module 488
13.3 Characteristic, Rank, and Height of Drinfeld Modules 490
13.4 Existence of Drinfeld Modules. Lattices 496
13.5 Explicit Class Field Theory 504
13.5.1 Class Number One Case 505
13.5.2 General Class Number Case 507
13.5.3 The Narrow Class Field //+ 512
13.5.4 The Hilbert Class Field HA 516
13.5.5 Explicit Class Fields and Ray Class Fields 518
13.6 Drinfeld Modules and Cryptography 521
13.6.1 Drinfeld Module Version of the Diffie Hellman Cryptosystem 522
13.6.2 The Gillard et al. Drinfeld Cryptosystem 522
13.7 Exercises 523
14 Automorphisms and Galois Theory 527
14.1 The Castelnuovo Severi Inequality 527
14.2 Weierstrass Points 532
14.2.1 Hasse Schmidt Differentials 534
14.2.2 The Wronskian 542
14.2.3 Arithmetic Theory of Weierstrass Points 551
14.2.4 Gap Sequences of Hyperelliptic Function Fields 561
14.2.5 Fields with Nonclassical Gap Sequence 566
14.3 Automorphism Groups of Algebraic Function Fields 570
14.4 Properties of Automorphisms of Function Fields 583
14.5 Exercises 593
A Cohomology of Groups 597
A. 1 Definitions and Basic Results 597
A.2 Homology and Cohomology in Low Dimensions 615
A.3 Tate Cohomology Groups 624
A.4 Cohomology of Cyclic Groups 627
A.5 Exercises 631
Notations 635
References 639
Index 647 |
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| index_date | 2024-07-02T17:05:13Z |
| indexdate | 2024-07-09T20:56:03Z |
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| series2 | Mathematics: theory & applications |
| spelling | Villa Salvador, Gabriel D. Verfasser (DE-588)132082713 aut Introducción a la teoría de las funciones algebraicas Topics in the theory of algebraic function fields Gabriel Daniel Villa Salvador Boston ; Basel ; Berlin Birkhäuser 2006 XVI, 652 S. 24 cm txt rdacontent n rdamedia nc rdacarrier Mathematics: theory & applications Literaturverz. S. 439 - 646 Presents an examination that explains both the similarities and fundamental differences between function fields and number fields, including many exercises and examples. This book serves as a text for a graduate course in number theory or an advanced graduate topics course. Mathematik Algebraic functions Functions Mathematics Algebraischer Funktionenkörper (DE-588)4141850-5 gnd rswk-swf Algebraischer Funktionenkörper (DE-588)4141850-5 s DE-604 text/html http://deposit.dnb.de/cgi-bin/dokserv?id=2748144&prov=M&dok_var=1&dok_ext=htm Inhaltstext HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015575133&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Villa Salvador, Gabriel D. Topics in the theory of algebraic function fields Mathematik Algebraic functions Functions Mathematics Algebraischer Funktionenkörper (DE-588)4141850-5 gnd |
| subject_GND | (DE-588)4141850-5 |
| title | Topics in the theory of algebraic function fields |
| title_alt | Introducción a la teoría de las funciones algebraicas |
| title_auth | Topics in the theory of algebraic function fields |
| title_exact_search | Topics in the theory of algebraic function fields |
| title_exact_search_txtP | Topics in the theory of algebraic function fields |
| title_full | Topics in the theory of algebraic function fields Gabriel Daniel Villa Salvador |
| title_fullStr | Topics in the theory of algebraic function fields Gabriel Daniel Villa Salvador |
| title_full_unstemmed | Topics in the theory of algebraic function fields Gabriel Daniel Villa Salvador |
| title_short | Topics in the theory of algebraic function fields |
| title_sort | topics in the theory of algebraic function fields |
| topic | Mathematik Algebraic functions Functions Mathematics Algebraischer Funktionenkörper (DE-588)4141850-5 gnd |
| topic_facet | Mathematik Algebraic functions Functions Mathematics Algebraischer Funktionenkörper |
| url | http://deposit.dnb.de/cgi-bin/dokserv?id=2748144&prov=M&dok_var=1&dok_ext=htm http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015575133&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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