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-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...
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"Very useful both for research and in the classroom. The main reason to use this book in a classroom is to prepare students for new research in the fields of finite geometries, curves in positive characteristic in a projective space, and curves over a finite field and their applications to coding theory. I think researchers will quote it for a long time."—Edoardo Ballico, University of Trento"This book is a self-contained guide to the theory of algebraic curves over a finite field, one that leads readers to various recent results in this and related areas. Personally I was attracted by the rich examples explained in this book."—Masaaki Homma, Kanagawa University Thomas Hagedorn - MAA Reviews 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.
Preface xiGeneral Theory of Curves 1Fundamental ideas 3Basic definitions 3Polynomials 6Affine plane curves 6Projective plane curves 9The Hessian curve 13Projective varieties in higher-dimensional spaces 18Exercises 18Notes 19Elimination theory 21Elimination of one unknown 21The discriminant 30Elimination in a system in two unknowns 31Exercises 35Notes 36Singular points and intersections 37The intersection number of two curves 37Bezout's Theorem 45Rational and birational transformations 49Quadratic transformations 51Resolution of singularities 55Exercises 61Notes 62Branches and parametrisation 63Formal power series 63Branch representations 75Branches of plane algebraic curves 81Local quadratic transformations 84Noether'sTheorem 92Analytic branches 99Exercises 107Notes 109The function field of a curve 110Generic points 110Rational transformations 112Places 119Zeros and poles 120Separability and inseparability 122Frobenius rational transformations 123Derivations and differentials 125The genus of a curve 130Residues of differential forms 138Higher derivatives in positive characteristic 144The dual and bidual of a curve 155Exercises 159Notes 160Linear series and the Riemann-Roch Theorem 161Divisors and linear series 161Linear systems of curves 170Special and non-special linear series 177Reformulation of the Riemann-Roch Theorem 180Some consequences of the Riemann-Roch Theorem 182The Weierstrass Gap Theorem 184The structure of the divisor class group 190Exercises 196Notes 198Algebraic curves in higher-dimensional spaces 199Basic definitions and properties 199Rational transformations 203Hurwitz's Theorem 208Linear series composed of an involution 211The canonical curve 216Osculating hyperplanes and ramification divisors 217Non-classical curves and linear systems of lines 228Non-classical curves and linear systems of conics 230Dual curves of space curves 238Complete linear series of small order 241Examples of curves 254The Linear General Position Principle 257Castelnuovo's Bound 257A generalisation of Clifford's Theorem 260The Uniform Position Principle 261Valuation rings 262Curves as algebraic varieties of dimension one 268Exercises 270Notes 271Curves over a Finite Field 275Rational points and places over a finite field 277Plane curves defined over a finite field 277F[subscript q]-rational branches of a curve 278F[subscript q]-rational places, divisors and linear series 281Space curves over F[subscript q] 287The Stohr-Voloch Theorem 292Frobenius classicality with respect to lines 305Frobenius classicality with respect to conics 314The dual of a Frobenius non-classical curve 326Exercises 327Notes 329Zeta functions and curves with many rational points 332The zeta function of a curve over a finite field 332The Hasse-Weil Theorem 343Refinements of the Hasse-Weil Theorem 348Asymptotic bounds 353Other estimates 356Counting points on a plane curve 358Further applications of the zeta function 369The Fundamental Equation 373Elliptic curves over F[subscript q] 378Classification of non-singular cubics over F[subscript q] 381Exercises 385Notes 388Further Developments 393Maximal and optimal curves 395Background on maximal curves 396The Frobenius linear series of a maximal curve 399Embedding in a Hermitian variety 407Maximal curves lying on a quadric surface 421Maximal curves with high genus 428Castelnuovo's number 431Plane maximal curves 439Maximal curves of Hurwitz type 442Non-isomorphic maximal curves 446Optimal curves 447Exercises 453Notes 454Automorphisms of an algebraic curve 458The action of K-automorphisms on places 459Linear series and automorphisms 464Automorphism groups of plane curves 468A bound on the order of a K-automorphism 470Automorphism groups and their fixed fields 473The stabiliser of a place 476Finiteness of the K-automorphism group 480Tame automorphism groups 483Non-tame automorphism groups 486K-automorphism groups of particular curves 501Fixed places of automorphisms 509Large automorphism groups of function fields 513K-automorphism groups fixing a place 532Large p-subgroups fixing a place 539Notes 542Some families of algebraic curves 546Plane curves given by separated polynomials 546Curves with Suzuki automorphism group 564Curves with unitary automorphism group 572Curves with Ree automorphism group 575A curve attaining the Serre Bound 585Notes 587Applications: codes and arcs 590Algebraic-geometry codes 590Maximum distance separable codes 594Arcs and ovals 599Segre's generalisation of Menelaus' Theorem 603The connection between arcs and curves 607Arcs in ovals in planes of even order 611Arcs in ovals in planes of odd order 612The second largest complete arc 615The third largest complete arc 623Exercises 625Notes 625Background on field theory and group theory 627Field theory 627Galois theory 633Norms and traces 635Finite fields 636Group theory 638Notes 649Notation 650Bibliography 655Index 689
\ MAA ReviewsThis 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.\ — Thomas Hagedorn\ \ \ \ \ MAA Reviews\ - Thomas Hagedorn\ 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.\ \
