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Algebraic Curves over a Finite Field

J. W. P. Hirschfeld, G. Korchmáros, and F. Torres
Publisher: 
Princeton University Press
Publication Date: 
2008
Number of Pages: 
696
Format: 
Hardcover
Series: 
Princeton Series in Applied Mathematics
Price: 
85.00
ISBN: 
9780691096797
Category: 
Monograph
[Reviewed by
Thomas Hagedorn
, on
07/2/2008
]

“Algebraic Curves over a Finite Field” is a rich, example-filled, comprehensive introduction to the subject. As an easy-to-read introductory book that presents the general theory of algebraic curves over finite fields, it fills a large gap in the literature.

Algebraic curves were the first examples (and the simplest) studied classically in algebraic geometry. Over a finite field, the theory is particularly rich, and the area has been a relatively accessible place where more general conjectures in algebraic geometry and number theory can be checked. For example, the analogue of the classical Riemann conjecture, the Riemann hypothesis for algebraic varieties over finite fields, was established for algebraic curves by Hasse and Weil in 1940, three decades before the general case was established by Dwork and Deligne.

Over the past decade, the field of algebraic curves over a finite field has been very active as the curves have found applications in number theory (construction of curves with a large number of points), coding theory (the construction of error-correcting codes), and finite geometry. This book brings the reader to the forefront of current research in these areas

The authors have done a remarkable job in presenting a concrete introduction to the subject. The book is divided into three parts. Part One presents an introduction to the general theory of algebraic curves. Part Two proves the key theorems in the subject: the Stohr-Voloch and Hasse-Weil (“Riemann Hypothesis”) theorems and the Serre bound. Part Three then surveys recent work in the field such as constructing curves with a maximal number of points or a given automorphism group

In 1991, Moreno’s almost identically titled book Algebraic Curves Over Finite Fields appeared; it is instructive to compare the two books. Both are self-contained and discuss a similar list of topics, but the current book has a more relaxed style (and so is longer: 696 pages versus 246 pages). The current book is filled with examples and explicit calculations that are not normally found in algebraic geometry textbooks.

Both books present the basic theory of algebraic curves and prove the Riemann hypothesis, but the second halves of the books differ. Moreno includes L-functions, estimates for exponential sums, and a chapter on Goppa curves and modular curves, whereas the emphasis of the second half of the current book is on finding optimal curves with a maximal number of points or a large automorphism group, with a short section on error-correcting codes and finite geometries concluding the book. Additionally, the current book has a more algebraic-geometric perspective, though the perspective is pre-Grothendieck (schemes do not appear)

This book is well-written and I greatly enjoyed reading it. The wealth of information and examples in this book give the reader a firm foundation and develop an intuition for the subject. The authors have used it as a textbook for a two-year course, and it would be a fine introduction to any advanced undergraduate or graduate student wanting to learn this subject. It would be a particularly good book for a student to read before tackling the standard introductory texts in algebraic geometry (Hartshorne, for example). Or, conversely, a student in an abstract algebraic geometry course could use this book to make the general theory more concrete.

Researchers in the subject will also find much useful information. This reviewer enjoyed the authors’ emphasis on non-classical curves that have properties that do not appear when working with real or complex curves. Also, introductions to topics such as dual curves are not easy to find in the literature. Their inclusion here is greatly welcomed. The bibliography and mathematical notes are also quite extensive and are useful pointers for someone looking to find research problems


Thomas Hagedorn is an Associate Professor of Mathematics and Statistics at The College of New Jersey. He can be reached at hagedorn at tcnj.edu.

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

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