Buch, Englisch, 716 Seiten, Format (B × H): 161 mm x 240 mm, Gewicht: 1230 g
Buch, Englisch, 716 Seiten, Format (B × H): 161 mm x 240 mm, Gewicht: 1230 g
Reihe: Princeton Series in Applied Mathematics
ISBN: 978-0-691-09679-7
Verlag: Princeton University Press
This book provides an accessible and self-contained introduction to the theory of algebraic curves over a finite field, a subject that has been of fundamental importance to mathematics for many years and that has essential applications in areas such as finite geometry, number theory, error-correcting codes, and cryptology. Unlike other books, this one emphasizes the algebraic geometry rather than the function field approach to algebraic curves. The authors begin by developing the general theory of curves over any field, highlighting peculiarities occurring for positive characteristic and requiring of the reader only basic knowledge of algebra and geometry. The special properties that a curve over a finite field can have are then discussed. The geometrical theory of linear series is used to find estimates for the number of rational points on a curve, following the theory of Stöhr and Voloch. The approach of Hasse and Weil via zeta functions is explained, and then attention turns to more advanced results: a state-of-the-art introduction to maximal curves over finite fields is provided; a comprehensive account is given of the automorphism group of a curve; and some applications to coding theory and finite geometry are described. The book includes many examples and exercises. It is an indispensable resource for researchers and the ideal textbook for graduate students.
Autoren/Hrsg.
Weitere Infos & Material
Preface xi
PART 1. GENERAL THEORY OF CURVES 1
Chapter 1. Fundamental ideas 3
1.1 Basic definitions 3
1.2 Polynomials 6
1.3 Affine plane curves 6
1.4 Projective plane curves 9
1.5 The Hessian curve 13
1.6 Projective varieties in higher-dimensional spaces 18
1.7 Exercises 18
1.8 Notes 19
Chapter 2. Elimination theory 21
2.1 Elimination of one unknown 21
2.2 The discriminant 30
2.3 Elimination in a system in two unknowns 31
2.4 Exercises 35
2.5 Notes 36
Chapter 3. Singular points and intersections 37
3.1 The intersection number of two curves 37
3.2 B?ezout's Theorem 45
3.3 Rational and birational transformations 49
3.4 Quadratic transformations 51
3.5 Resolution of singularities 55
3.6 Exercises 61
3.7 Notes 62
Chapter 4. Branches and parametrisation 63
4.1 Formal power series 63
4.2 Branch representations 75
4.3 Branches of plane algebraic curves 81
4.4 Local quadratic transformations 84
4.5 Noether's Theorem 92
4.6 Analytic branches 99
4.7 Exercises 107
4.8 Notes 109
Chapter 5. The function field of a curve 110
5.1 Generic points 110
5.2 Rational transformations 112
5.3 Places 119
5.4 Zeros and poles 120
5.5 Separability and inseparability 122
5.6 Frobenius rational transformations 123
5.7 Derivations and differentials 125
5.8 The genus of a curve 130
5.9 Residues of differential forms 138
5.10 Higher derivatives in positive characteristic 144
5.11 The dual and bidual of a curve 155
5.12 Exercises 159
5.13 Notes 160
Chapter 6. Linear series and the Riemann-Roch Theorem 161
6.1 Divisors and linear series 161
6.2 Linear systems of curves 170
6.3 Special and non-special linear series 177
6.4 Reformulation of the Riemann-Roch Theorem 180
6.5 Some consequences of the Riemann-Roch Theorem 182
6.6 The Weierstrass Gap Theorem 184
6.7 The structure of the divisor class group 190
6.8 Exercises 196
6.9 Notes 198
Chapter 7. Algebraic curves in higher-dimensional spaces 199
7.1 Basic definitions and properties 199
7.2 Rational transformations 203
7.3 Hurwitz's Theorem 208
7.4 Linear series composed of an involution 211
7.5 The canonical curve 216
7.6 Osculating hyperplanes and ramification divisors 217
7.7 Non-classical curves and linear systems of lines 228
7.8 Non-classical curves and linear systems of conics 230
7.9 Dual curves of space curves 238
7.10 Complete linear series of small order 241
7.11 Examples of curves 254
7.12 The Linear General Position Principle 257
7.13 Castelnuovo's Bound 257
7.14 A generalisation of Clifford's Theorem 260
7.15 The Uniform Position Principle 261
7.16 Valuation rings 262
7.17 Curves as algebraic varieties of dimension one 268
7.18 Exercises 270
7.19 Notes 271
PART 2. CURVES OVER A FINITE FIELD 275
Chapter 8. Rational points and places over a finite field 277
8.1 Plane curves defined over a finite field 277
8.2 Fq-rational branches of a curve 278
8.3 Fq-rational places, divisors and linear series 281
8.4 Space curves over Fq 287
8.5 The St?ohr-Voloch Theorem 292
8.6 Frobenius classicality with respect to lines 305
8.7 Frobenius classicality with respect to conics 314
8.8 The dual of a Frobenius non-classical curve 326
8.9 Exercises 327
8.10 Notes 329
Chapter 9. Zeta functions and curves with many rational points 332
9.1 The zeta function of a curve over a finite field 332
9.2 The Hasse-Weil Theorem 343
9.3 Refinements of the Hasse-Weil Theorem 348
9.4 Asymptotic bounds 353
9.5 Other estimates 356
9.6 Counting points on a plane curve 358
9.7 Further applications of the zeta function 369
9.8 The Fundamental Equation 373
9.9 Elliptic curves over Fq 378
9.10 Classification of non-singular cubics over Fq 381
9.11 Exercises 385
9.12 Notes 388
PART 3. FURTHER DEVELOPMENTS 393
Chapter 10. Maximal
