Topics in the Theory of Algebraic Function Fields (Mathematics: Theory & Applications)

Preface......Page 8
Contents......Page 14
1.1 Algebraic and Transcendental Extensions......Page 19
1.2 Absolute Values over......Page 21
1.3 Riemann Surfaces......Page 26
1.4 Exercises......Page 29
2 Algebraic Function Fields of One Variable......Page 31
2.1 The Field of Constants......Page 32
2.2 Valuations, Places, and Valuation Rings......Page 34
2.3 Absolute Values and Completions......Page 44
2.4 Valuations in Rational Function Fields......Page 54
2.5 Artin s Approximation Theorem......Page 61
2.6 Exercises......Page 70
3.1 Divisors......Page 73
3.2 Principal Divisors and Class Groups......Page 79
3.3 Repartitions or Adeles......Page 85
3.4 Differentials......Page 90
3.5 The Riemann…Roch Theorem and Its Applications......Page 99
3.6 Exercises......Page 106
4.1 Fields of Rational Functions and Function Fields of Genus......Page 111
4.2 Elliptic Function Fields and Function Fields of Genus......Page 119
4.3 Quadratic Extensions of k(x) and Computation of the Genus......Page 123
4.4 Exercises......Page 129
5.1 Extensions of Function Fields......Page 131
5.2 Galois Extensions of Function Fields......Page 136
5.3 Divisors in an Extension......Page 146
5.4 Completions and Galois Theory......Page 150
5.5 Integral Bases......Page 156
5.6 Different and Discriminant......Page 165
5.7 Dedekind Domains......Page 168
5.8 Rami“cation in Artin…Schreier and Kummer Extensions......Page 182
5.9 Rami“cation Groups......Page 198
5.10 Exercises......Page 204
6.1 Constant Extensions......Page 209
6.2 Prime Divisos in Cnstant Extensins......Page 211
6.3 Zeta Functions and L-Series......Page 213
6.4 Functional Equations......Page 218
6.5 Exercises......Page 225
7.1 The Number of Prime Divisors of Degree......Page 227
7.2 Proof of the Riemann hypothesis......Page 233
7.3 Consequences of the Riemann Hypothesis......Page 240
7.4 Function Fields with Small Class Number......Page 245
7.5 The Class Numbers of Congruence Function Fields......Page 249
7.6 The Analogue of the Brauer…Siegel Theorem......Page 252
7.7 Exercises......Page 255
8.1 Linearly Disjoint Extensions......Page 257
8.2 Separable and Separably Generated Extensions......Page 262
8.3 Regular Extensions......Page 268
8.4 Constant Extensions......Page 271
8.5 Genus Change in Constant Extensions......Page 283
8.6 Inseparable Function Fields......Page 294
8.7 Exercises......Page 299
9.1 The Differential dx in........Page 301
9.2 Trace and Cotrace of Differentials......Page 307
9.3 Hasse Differentials and Residues......Page 310
9.4 The Genus Formula......Page 325
9.5 Genus Change in Inseparable Extensions......Page 329
9.6 Examples......Page 343
9.7 Exercises......Page 369
10.1 Introduction......Page 371
10.2 Symmetric and Asymmetric Cryptosystems......Page 372
10.3 Finite Field Cryptosystems......Page 374
10.4 Elliptic Function Fields Cryptosystems......Page 376
10.5 The ElGamal Cryptosystem......Page 378
10.6 Hyperelliptic Cryptosystems......Page 381
10.7 Reduced Divisors over Finite Fields......Page 385
10.8 Implementation of Hyperelliptic Cryptosystems......Page 388
10.9 Exercises......Page 392
11.1 Introduction......Page 395
11.2 Cebotar ev s Density Theorem......Page 396
11.3 Inverse Limits and Pro“nite Groups......Page 406
11.4 In“nite Galois Theory......Page 418
11.5 Results on Global Class Field Theory......Page 427
11.7 Exercises......Page 429
12.1 Introduction......Page 433
12.2 Basic Facts......Page 434
12.3 Cyclotomic Function Fields......Page 440
12.4 Arithmetic of Cyclotomic Function Fields......Page 447
12.5 The Artin Symbol in Cyclotomic Function Fields......Page 456
12.6 Dirichlet Characters......Page 466
12.7 Different and Genus......Page 479
12.8 The Maximal Abelian Extension of......Page 481
12.9 The Analogue of the Brauer…Siegel Theorem......Page 496
12.10 Exercises......Page 498
13.1 Introduction......Page 505
13.2 Additive Polynomials and the Carlitz Module......Page 506
13.3 Characteristic, Rank, and Height of Drinfeld Modules......Page 508
13.4 Existence of Drinfeld Modules. Lattices......Page 514
13.5 Explicit Class Field Theory......Page 522
13.6 Drinfeld Modules and Cryptography......Page 539
13.7 Exercises......Page 541
14.1 The Castelnuovo…Severi Inequality......Page 545
14.2 Weierstrass Points......Page 550
14.3 Automorphism Groups of Algebraic Function Fields......Page 588
14.4 Properties of Automorphisms of Function Fields......Page 601
14.5 Exercises......Page 611
A.1 De“nitions and Basic Results......Page 615
A.2 Homology and Cohomology in Low Dimensions......Page 633
A.3 Tate Cohomology Groups......Page 642
A.4 Cohomology of Cyclic Groups......Page 645
A.5 Exercises......Page 649
Notations......Page 653
References......Page 657
Index......Page 665