Algebraic curves over a finite field

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-correctin...

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Bibliographic Details
Main Authors:	Hirschfeld, J. W. P, Korchmaros, Gabor, Torres, Fernando
Format:	eBook Book
Language:	English
Published:	Princeton, N.J Princeton University Press 2013
Edition:	STU - Student edition
Subjects:	Affine plane Affine space Affine variety Algebraic closure Algebraic curve Algebraic equation Algebraic extension Algebraic function Algebraic geometry Algebraic integer Algebraic number Algebraic number field Algebraic number theory Algebraic variety Algebraically closed field Automorphism Birational invariant Classification theorem Clifford's theorem Combinatorics Curves, Algebraic Cyclotomic polynomial Degeneracy (mathematics) Divisor Divisor (algebraic geometry) Dual curve Elliptic curve Equation Finite field Finite fields (Algebra) Finite geometry Function (mathematics) Function field Galois extension Galois theory Gauss map Generic point Geometry Hurwitz's theorem Hyperelliptic curve Hyperplane Intersection number (graph theory) J-invariant Line at infinity Linear map Mathematical induction MATHEMATICS MATHEMATICS / Algebra / General MATHEMATICS / Applied Menelaus' theorem Modular curve Number theory Parity (mathematics) Permutation group Plane curve Point at infinity Polar curve Polynomial Projective plane Projective space Quadratic transformation Resolution of singularities Riemann hypothesis Scalar multiplication Separable extension Separable polynomial Sign (mathematics) Subgroup Sylow theorems Theorem Transcendence degree Valuation ring Variable (mathematics) Vector space Algebraic function Polynomial Quadratic transformation Separable extension J-invariant Subgroup Algebraic integer Birational invariant Divisor (algebraic geometry) Vector space Mathematical induction Projective plane Degeneracy (mathematics) Algebraic curve Combinatorics Algebraic number theory Finite field Polar curve Intersection number (graph theory) Line at infinity Cyclotomic polynomial Function (mathematics) Hyperplane Gauss map Transcendence degree Plane curve Permutation group Riemann hypothesis Projective space Sign (mathematics) Algebraically closed field Point at infinity Mathematics Affine variety Function field Linear map Algebraic extension Sylow theorems Hyperelliptic curve Theorem Valuation ring Hurwitz's theorem Automorphism Generic point Affine plane Menelaus' theorem Equation Elliptic curve Dual curve Galois theory Modular curve Affine space Algebraic number field Variable (mathematics) Geometry Clifford's theorem Algebraic variety Resolution of singularities Finite geometry Parity (mathematics) Algebraic equation Divisor Galois extension Algebraic closure Algebraic number Separable polynomial Classification theorem Algebraic geometry Scalar multiplication Number theory
ISBN:	9781400847419, 1400847419, 9780691096797, 0691096791
Online Access:	Get full text
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Table of Contents:

Algebraic curves over a finite field -- Contents -- Preface -- Part 1: General Theory of Curves -- Chapter 1: Fundamental Ideas -- Chapter 2: Elimination Theory -- Chapter 3: Singular Points and Intersections -- Chapter 4: Branches and Parametrisation -- Chapter 5: The Function Field of a Curve -- Chapter 6: Linear Series and the Riemann–Roch Theorem -- Chapter 7: Algebraic Curves in Higher-Dimensional Spaces -- Part 2: Curves Over a Finite Field -- Chapter 8: Rational Points and Places Over a Finite Field -- Chapter 9: Zeta Functions and Curves with Many Rational Points -- Part 3: Further Developments -- Chapter 10: Maximal and Optimal Curves -- Chapter 11: Automorphisms of an Algebraic Curve -- Chapter 12: Some Families of Algebraic Curves -- Chapter 13: Applications: Codes and Arcs -- Appendix A: Background on Field Theory and Group Theory -- Appendix B: Notation -- Bibliography -- Index
Front Matter Table of Contents Preface Chapter One: Fundamental ideas Chapter Two: Elimination theory Chapter Three: Singular points and intersections Chapter Four: Branches and parametrisation Chapter Five: The function field of a curve Chapter Six: Linear series and the Riemann–Roch Theorem Chapter Seven: Algebraic curves in higher-dimensional spaces Chapter Eight: Rational points and places over a finite field Chapter Nine: Zeta functions and curves with many rational points Chapter Ten: Maximal and optimal curves Chapter Eleven: Automorphisms of an algebraic curve Chapter Twelve: Some families of algebraic curves Chapter Thirteen: Applications: Appendix A. Appendix B. Bibliography Index
A.2 Galois theory -- A.3 Norms and traces -- A.4 Finite fields -- A.5 Group theory -- A.6 Notes -- Appendix B. Notation -- Bibliography -- Index
10.2 The Frobenius linear series of a maximal curve -- 10.3 Embedding in a Hermitian variety -- 10.4 Maximal curves lying on a quadric surface -- 10.5 Maximal curves with high genus -- 10.6 Castelnuovo's number -- 10.7 Plane maximal curves -- 10.8 Maximal curves of Hurwitz type -- 10.9 Non-isomorphic maximal curves -- 10.10 Optimal curves -- 10.11 Exercises -- 10.12 Notes -- Chapter 11. Automorphisms of an algebraic curve -- 11.1 The action of K-automorphisms on places -- 11.2 Linear series and automorphisms -- 11.3 Automorphism groups of plane curves -- 11.4 A bound on the order of a K-automorphism -- 11.5 Automorphism groups and their fixed fields -- 11.6 The stabiliser of a place -- 11.7 Finiteness of the K-automorphism group -- 11.8 Tame automorphism groups -- 11.9 Non-tame automorphism groups -- 11.10 K-automorphism groups of particular curves -- 11.11 Fixed places of automorphisms -- 11.12 Large automorphism groups of function fields -- 11.13 K-automorphism groups fixing a place -- 11.14 Large p-subgroups fixing a place -- 11.15 Notes -- Chapter 12. Some families of algebraic curves -- 12.1 Plane curves given by separated polynomials -- 12.2 Curves with Suzuki automorphism group -- 12.3 Curves with unitary automorphism group -- 12.4 Curves with Ree automorphism group -- 12.5 A curve attaining the Serre Bound -- 12.6 Notes -- Chapter 13. Applications: codes and arcs -- 13.1 Algebraic-geometry codes -- 13.2 Maximum distance separable codes -- 13.3 Arcs and ovals -- 13.4 Segre's generalisation of Menelaus' Theorem -- 13.5 The connection between arcs and curves -- 13.6 Arcs in ovals in planes of even order -- 13.7 Arcs in ovals in planes of odd order -- 13.8 The second largest complete arc -- 13.9 The third largest complete arc -- 13.10 Exercises -- 13.11 Notes -- Appendix A. Background on field theory and group theory -- A.1 Field theory
Cover -- Title -- Copyright -- Dedication -- Contents -- Preface -- PART 1. GENERAL THEORY OF CURVES -- Chapter 1. Fundamental ideas -- 1.1 Basic definitions -- 1.2 Polynomials -- 1.3 Affine plane curves -- 1.4 Projective plane curves -- 1.5 The Hessian curve -- 1.6 Projective varieties in higher-dimensional spaces -- 1.7 Exercises -- 1.8 Notes -- Chapter 2. Elimination theory -- 2.1 Elimination of one unknown -- 2.2 The discriminant -- 2.3 Elimination in a system in two unknowns -- 2.4 Exercises -- 2.5 Notes -- Chapter 3. Singular points and intersections -- 3.1 The intersection number of two curves -- 3.2 Bézout's Theorem -- 3.3 Rational and birational transformations -- 3.4 Quadratic transformations -- 3.5 Resolution of singularities -- 3.6 Exercises -- 3.7 Notes -- Chapter 4. Branches and parametrisation -- 4.1 Formal power series -- 4.2 Branch representations -- 4.3 Branches of plane algebraic curves -- 4.4 Local quadratic transformations -- 4.5 Noether's Theorem -- 4.6 Analytic branches -- 4.7 Exercises -- 4.8 Notes -- Chapter 5. The function field of a curve -- 5.1 Generic points -- 5.2 Rational transformations -- 5.3 Places -- 5.4 Zeros and poles -- 5.5 Separability and inseparability -- 5.6 Frobenius rational transformations -- 5.7 Derivations and differentials -- 5.8 The genus of a curve -- 5.9 Residues of differential forms -- 5.10 Higher derivatives in positive characteristic -- 5.11 The dual and bidual of a curve -- 5.12 Exercises -- 5.13 Notes -- Chapter 6. Linear series and the Riemann-Roch Theorem -- 6.1 Divisors and linear series -- 6.2 Linear systems of curves -- 6.3 Special and non-special linear series -- 6.4 Reformulation of the Riemann-Roch Theorem -- 6.5 Some consequences of the Riemann-Roch Theorem -- 6.6 The Weierstrass Gap Theorem -- 6.7 The structure of the divisor class group -- 6.8 Exercises -- 6.9 Notes
Chapter 7. Algebraic curves in higher-dimensional spaces -- 7.1 Basic definitions and properties -- 7.2 Rational transformations -- 7.3 Hurwitz's Theorem -- 7.4 Linear series composed of an involution -- 7.5 The canonical curve -- 7.6 Osculating hyperplanes and ramification divisors -- 7.7 Non-classical curves and linear systems of lines -- 7.8 Non-classical curves and linear systems of conics -- 7.9 Dual curves of space curves -- 7.10 Complete linear series of small order -- 7.11 Examples of curves -- 7.12 The Linear General Position Principle -- 7.13 Castelnuovo's Bound -- 7.14 A generalisation of Clifford's Theorem -- 7.15 The Uniform Position Principle -- 7.16 Valuation rings -- 7.17 Curves as algebraic varieties of dimension one -- 7.18 Exercises -- 7.19 Notes -- PART 2. CURVES OVER A FINITE FIELD -- Chapter 8. Rational points and places over a finite field -- 8.1 Plane curves defined over a finite field -- 8.2 Fq-rational branches of a curve -- 8.3 Fq-rational places, divisors and linear series -- 8.4 Space curves over Fq -- 8.5 The Stöhr-Voloch Theorem -- 8.6 Frobenius classicality with respect to lines -- 8.7 Frobenius classicality with respect to conics -- 8.8 The dual of a Frobenius non-classical curve -- 8.9 Exercises -- 8.10 Notes -- Chapter 9. Zeta functions and curves with many rational points -- 9.1 The zeta function of a curve over a finite field -- 9.2 The Hasse-Weil Theorem -- 9.3 Refinements of the Hasse-Weil Theorem -- 9.4 Asymptotic bounds -- 9.5 Other estimates -- 9.6 Counting points on a plane curve -- 9.7 Further applications of the zeta function -- 9.8 The Fundamental Equation -- 9.9 Elliptic curves over Fq -- 9.10 Classification of non-singular cubics over Fq -- 9.11 Exercises -- 9.12 Notes -- PART 3. FURTHER DEVELOPMENTS -- Chapter 10. Maximal and optimal curves -- 10.1 Background on maximal curves
Chapter Eleven. Automorphisms of an algebraic curve
Appendix B. Notation
Chapter Twelve. Some families of algebraic curves
Chapter Eight. Rational points and places over a finite field
Index
Chapter Five. The function field of a curve
Chapter Two. Elimination theory
Chapter Six. Linear series and the Riemann–Roch Theorem
Preface
PART 1. General theory of curves --
PART 3. Further developments --
PART 2. Curves over a finite field --
Appendix A. Background on field theory and group theory
Chapter Three. Singular points and intersections
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Chapter Thirteen. Applications: codes and arcs
Chapter Four. Branches and parametrisation
Chapter Nine. Zeta functions and curves with many rational points
Contents
Chapter One. Fundamental ideas
Chapter Seven. Algebraic curves in higher-dimensional spaces
Frontmatter --
Bibliography
Chapter Ten. Maximal and optimal curves