Algebraic Curves over a Finite Field

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 and optimal curves 395
10.1 Background on maximal curves 396
10.2 The Frobenius linear series of a maximal curve 399
10.3 Embedding in a Hermitian variety 407
10.4 Maximal curves lying on a quadric surface 421
10.5 Maximal curves with high genus 428
10.6 Castelnuovo's number 431
10.7 Plane maximal curves 439
10.8 Maximal curves of Hurwitz type 442
10.9 Non-isomorphic maximal curves 446
10.10 Optimal curves 447
10.11 Exercises 453
10.12 Notes 454
Chapter 11. Automorphisms of an algebraic curve 458
11.1 The action of K-automorphisms on places 459
11.2 Linear series and automorphisms 464
11.3 Automorphism groups of plane curves 468
11.4 A bound on the order of a K-automorphism 470
11.5 Automorphism groups and their fixed fields 473
11.6 The stabiliser of a place 476
11.7 Finiteness of the K-automorphism group 480
11.8 Tame automorphism groups 483
11.9 Non-tame automorphism groups 486
11.10 K-automorphism groups of particular curves 501
11.11 Fixed places of automorphisms 509
11.12 Large automorphism groups of function fields 513
11.13 K-automorphism groups fixing a place 532
11.14 Large p-subgroups fixing a place 539
11.15 Notes 542
Chapter 12. Some families of algebraic curves 546
12.1 Plane curves given by separated polynomials 546
12.2 Curves with Suzuki automorphism group 564
12.3 Curves with unitary automorphism group 572
12.4 Curves with Ree automorphism group 575
12.5 A curve attaining the Serre Bound 585
12.6 Notes 587
Chapter 13. Applications: codes and arcs 590
13.1 Algebraic-geometry codes 590
13.2 Maximum distance separable codes 594
13.3 Arcs and ovals 599
13.4 Segre's generalisation of Menelaus' Theorem 603
13.5 The connection between arcs and curves 607
13.6 Arcs in ovals in planes of even order 611
13.7 Arcs in ovals in planes of odd order 612
13.8 The second largest complete arc 615
13.9 The third largest complete arc 623
13.10 Exercises 625
13.11 Notes 625

Appendix A. Background on field theory and group theory 627
A.1 Field theory 627
A.2 Galois theory 633
A.3 Norms and traces 635
A.4 Finite fields 636
A.5 Group theory 638
A.6 Notes 649
Appendix B. Notation 650
Bibliography 655
Index 689