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# Algebraic Curves over a Finite FieldJ.W.P. Hirschfeld, G. Korchmáros & F. Torres

 TABLE OF CONTENTS:Preface xi PART 1. GENERAL THEORY OF CURVES 1Chapter 1. Fundamental ideas 31.1 Basic definitions 31.2 Polynomials 61.3 Affine plane curves 61.4 Projective plane curves 91.5 The Hessian curve 131.6 Projective varieties in higher-dimensional spaces 181.7 Exercises 181.8 Notes 19Chapter 2. Elimination theory 212.1 Elimination of one unknown 212.2 The discriminant 302.3 Elimination in a system in two unknowns 312.4 Exercises 352.5 Notes 36Chapter 3. Singular points and intersections 373.1 The intersection number of two curves 373.2 B´ezout's Theorem 453.3 Rational and birational transformations 493.4 Quadratic transformations 513.5 Resolution of singularities 553.6 Exercises 613.7 Notes 62Chapter 4. Branches and parametrisation 634.1 Formal power series 634.2 Branch representations 754.3 Branches of plane algebraic curves 814.4 Local quadratic transformations 844.5 Noether's Theorem 924.6 Analytic branches 994.7 Exercises 1074.8 Notes 109Chapter 5. The function field of a curve 1105.1 Generic points 1105.2 Rational transformations 1125.3 Places 1195.4 Zeros and poles 1205.5 Separability and inseparability 1225.6 Frobenius rational transformations 1235.7 Derivations and differentials 1255.8 The genus of a curve 1305.9 Residues of differential forms 1385.10 Higher derivatives in positive characteristic 1445.11 The dual and bidual of a curve 1555.12 Exercises 1595.13 Notes 160Chapter 6. Linear series and the Riemann-Roch Theorem 1616.1 Divisors and linear series 1616.2 Linear systems of curves 1706.3 Special and non-special linear series 1776.4 Reformulation of the Riemann-Roch Theorem 1806.5 Some consequences of the Riemann-Roch Theorem 1826.6 The Weierstrass Gap Theorem 1846.7 The structure of the divisor class group 1906.8 Exercises 1966.9 Notes 198Chapter 7. Algebraic curves in higher-dimensional spaces 1997.1 Basic definitions and properties 1997.2 Rational transformations 2037.3 Hurwitz's Theorem 2087.4 Linear series composed of an involution 2117.5 The canonical curve 2167.6 Osculating hyperplanes and ramification divisors 2177.7 Non-classical curves and linear systems of lines 2287.8 Non-classical curves and linear systems of conics 2307.9 Dual curves of space curves 2387.10 Complete linear series of small order 2417.11 Examples of curves 2547.12 The Linear General Position Principle 2577.13 Castelnuovo's Bound 2577.14 A generalisation of Clifford's Theorem 2607.15 The Uniform Position Principle 2617.16 Valuation rings 2627.17 Curves as algebraic varieties of dimension one 2687.18 Exercises 2707.19 Notes 271PART 2. CURVES OVER A FINITE FIELD 275Chapter 8. Rational points and places over a finite field 2778.1 Plane curves defined over a finite field 2778.2 Fq-rational branches of a curve 2788.3 Fq-rational places, divisors and linear series 2818.4 Space curves over Fq 2878.5 The St¨ohr-Voloch Theorem 2928.6 Frobenius classicality with respect to lines 3058.7 Frobenius classicality with respect to conics 3148.8 The dual of a Frobenius non-classical curve 3268.9 Exercises 3278.10 Notes 329Chapter 9. Zeta functions and curves with many rational points 3329.1 The zeta function of a curve over a finite field 3329.2 The Hasse-Weil Theorem 3439.3 Refinements of the Hasse-Weil Theorem 3489.4 Asymptotic bounds 3539.5 Other estimates 3569.6 Counting points on a plane curve 3589.7 Further applications of the zeta function 3699.8 The Fundamental Equation 3739.9 Elliptic curves over Fq 3789.10 Classification of non-singular cubics over Fq 3819.11 Exercises 3859.12 Notes 388PART 3. FURTHER DEVELOPMENTS 393Chapter 10. Maximal and optimal curves 39510.1 Background on maximal curves 39610.2 The Frobenius linear series of a maximal curve 39910.3 Embedding in a Hermitian variety 40710.4 Maximal curves lying on a quadric surface 42110.5 Maximal curves with high genus 42810.6 Castelnuovo's number 43110.7 Plane maximal curves 43910.8 Maximal curves of Hurwitz type 44210.9 Non-isomorphic maximal curves 44610.10 Optimal curves 44710.11 Exercises 45310.12 Notes 454Chapter 11. Automorphisms of an algebraic curve 45811.1 The action of K-automorphisms on places 45911.2 Linear series and automorphisms 46411.3 Automorphism groups of plane curves 46811.4 A bound on the order of a K-automorphism 47011.5 Automorphism groups and their fixed fields 47311.6 The stabiliser of a place 47611.7 Finiteness of the K-automorphism group 48011.8 Tame automorphism groups 48311.9 Non-tame automorphism groups 48611.10 K-automorphism groups of particular curves 50111.11 Fixed places of automorphisms 50911.12 Large automorphism groups of function fields 51311.13 K-automorphism groups fixing a place 53211.14 Large p-subgroups fixing a place 53911.15 Notes 542Chapter 12. Some families of algebraic curves 54612.1 Plane curves given by separated polynomials 54612.2 Curves with Suzuki automorphism group 56412.3 Curves with unitary automorphism group 57212.4 Curves with Ree automorphism group 57512.5 A curve attaining the Serre Bound 58512.6 Notes 587Chapter 13. Applications: codes and arcs 59013.1 Algebraic-geometry codes 59013.2 Maximum distance separable codes 59413.3 Arcs and ovals 59913.4 Segre's generalisation of Menelaus' Theorem 60313.5 The connection between arcs and curves 60713.6 Arcs in ovals in planes of even order 61113.7 Arcs in ovals in planes of odd order 61213.8 The second largest complete arc 61513.9 The third largest complete arc 62313.10 Exercises 62513.11 Notes 625Appendix A. Background on field theory and group theory 627A.1 Field theory 627A.2 Galois theory 633A.3 Norms and traces 635A.4 Finite fields 636A.5 Group theory 638A.6 Notes 649Appendix B. Notation 650Bibliography 655Index 689Return to Book DescriptionFile created: 4/21/2017 Questions and comments to: webmaster@press.princeton.eduPrinceton University Press