Algebraic Curves and Riemann Surfaces for Undergraduates: The Theory of the Donut

Description

The theory relating algebraic curves and Riemann surfaces exhibits the unity of mathematics: topology, complex analysis, algebra and geometry all interact in a deep way. This textbook offers an elementary introduction to this beautiful theory for an undergraduate audience. At the heart of the subject is the theory of elliptic functions and elliptic curves. A complex torus (or “donut”) is both an abelian group and a Riemann surface. It is obtained by identifying points on the complex plane. At the same time, it can be viewed as a complex algebraic curve, with addition of points given by a geometric “chord-and-tangent” method. This book carefully develops all of the tools necessary to make sense of this isomorphism. The exposition is kept as elementary as possible and frequently draws on familiar notions in calculus and algebra to motivate new concepts. Based on a capstone course given to senior undergraduates, this book is intended as a textbook for courses at this level and includes a large number of class-tested exercises. The prerequisites for using the book are familiarity with abstract algebra, calculus and analysis, as covered in standard undergraduate courses. Preface Contents List of Symbols 1 Introduction 1.1 The Theory of the Circle 1.1.1 Pythagorean Triples 1.1.2 The Circular Functions The Isomorphism Theorem for the Circle 1.1.3 The Theory of the Donut, in a Nutshell 1.2 Overview of the Book 1.2.1 Part I: Algebraic Curves Affine and Projective Curves Intersections of Curves Elliptic Curves 1.2.2 Part II: Riemann Surfaces Three Kinds of Surfaces Analytic Functions Real Analysis, Complex Analysis, and Path Integrals Finally, Riemann Surfaces 1.2.3 Part III: Curves and Surfaces 1.3 Preliminaries, and Some Notation Part I Algebraic Curves 2 Algebra 2.1 Polynomials and Power Series 2.1.1 The Category of Rings Simplifying Notation Derived Properties 2.1.2 Back to Formal Power Series Infinite Sums Several Variables 2.1.3 More on Polynomials The Degree of a Polynomial Polynomial Substitution 2.2 Unique Factorisation 2.2.1 Divisibility in Integral Domains Divisibility in R[x] 2.2.2 Unique Factorisation Domains Multisets Unique Factorisation 2.2.3 Unique Factorisation in Polynomial Rings One Variable Interlude: Algebraically Closed Fields Gauss's Lemmas 2.3 Groups 2.3.1 The Category of Groups Subgroups Group Homomorphisms 2.3.2 Quotient Groups 2.3.3 Cyclic Groups The Characteristic of a Ring 2.3.4 The Symmetric Group 2.4 Linear Algebra Over Integral Domains 2.4.1 Matrices, Linear Spaces, and Linear Maps Linear Spaces Invertible and Nonsingular Matrices 2.4.2 Dimension and Complements 2.4.3 The Determinant The Effect of Row Operations Polynomial Substitution 2.4.4 Detecting Singularity 2.5 Further Exercises 3 Affine Space 3.1 Definition of Hypersurfaces 3.2 The Resultant 3.2.1 The Sylvester Matrix 3.2.2 The Resultant, Common Roots, and More Variables Adding More Variables 3.2.3 The Resultant is a Linear Combination 3.3 Study's Lemma 3.3.1 Proof of Study's Lemma 3.4 Affine Lines and Rational Parameterisations 3.4.1 Affine Lines 3.4.2 Rational Parameterisations 3.5 Further Exercises 4 Projective Space 4.1 Homogeneous Polynomials A Characterisation of Homogeneity 4.2 Projective Space 4.3 Projective Lines and Maps 4.3.1 Projective Maps 4.4 Embedding Affine Space into Projective Space Vertical and Horizontal Projective Lines Affine Cover and the Riemann Sphere Algebraic Subsets of the Projective Line 4.5 Changes of Coordinates 4.5.1 Change of Variable 4.5.2 Four Point Lemma 4.6 Spaces of Curves 4.6.1 The Dual Plane Principle of Duality 4.6.2 Desargues' Theorem 4.7 Products of Projective Spaces 4.8 Further Exercises 5 Tangents 5.1 Introduction: Affine Tangents and Intersections with Lines 5.1.1 Intersection Multiplicities 5.1.2 Homogeneous Coordinates 5.2 Formal Partial Derivatives 5.2.1 Properties of Derivatives The Chain Rule Euler's Relation Taylor Expansions 5.2.2 The Discriminant 5.3 Higher Order Tangents The Affine Higher Order Tangent 5.3.1 The Moduli Space of Tangents 5.3.2 Invariance of the Higher Order Tangent 5.4 The Intersection of a Line with a Curve 5.4.1 Definition of Intersection Multiplicity Bézout for a Line 5.4.2 Invariance of Multiplicity of Intersection with a Line Affine Calculations 5.4.3 Tangents and Intersections with Lines Defining Multiplicities Using Tangents 5.4.4 Simple Intersections Are the Norm 5.5 Further Exercises 6 Bézout's Theorem 6.1 A First Look at the Intersection of Curves 6.1.1 The Resultant of Homogeneous Polynomials Is Homogeneous 6.1.2 A Weak Version of Bézout's Theorem A Naïve Definition of Intersection Multiplicity 6.2 The Homogeneous Resultant 6.2.1 Main Property of the Homogeneous Resultant 6.3 Multiplicity of Intersection and Bézout's Theorem 6.3.1 Coding Lines in ps: [/EMC pdfmark [/Subtype /Span /ActualText (double struck upper P squared times double struck upper P squared) /StPNE pdfmark [/StBMC pdfmarkP2P2ps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark 6.3.2 The Resultant of the General Intersection Polynomials Bihomogeneity of Rf,g The Structure of the Hypersurface Defined by Rf,g 6.3.3 Intersection Multiplicity and Bézout's Theorem The Intersection Multiset 6.3.4 Geometric Invariance 6.4 Coincidence with Earlier Definitions 6.4.1 Using the Family of Vertical Lines 6.4.2 Intersecting Lines 6.5 Categoricity of Multiplicity of Intersection 6.5.1 Symmetry 6.5.2 Products 6.5.3 Infinite Multiplicities 6.5.4 Shifts 6.5.5 Categoricity of Multiplicity of Intersection 6.6 Affine Calculations 6.7 Multiplicities, Orders and Tangents 6.8 Further Exercises 7 The Elliptic Group 7.1 Flexes 7.1.1 Flexes and the Second Order Tangent 7.1.2 The Hessian 7.2 The Group Operation on a Nonsingular Cubic Curve 7.2.1 The Complement Curve 7.2.2 Associativity of the Group Operation 7.3 Normal Forms for Nonsingular Cubics 7.3.1 Explicit Calculations of the Group Operation 7.4 Further Exercises Part II Riemann Surfaces 8 Quasi-Euclidean Spaces 8.1 Topology of ps: [/EMC pdfmark [/Subtype /Span /ActualText (double struck upper R Superscript n) /StPNE pdfmark [/StBMC pdfmarkRnps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark 8.2 Manifolds Examples 8.2.1 Topology of Pre-manifolds 8.2.2 Subspaces 8.2.3 The Hausdorff Property 8.2.4 Topological Countability 8.2.5 Manifolds 8.2.6 Spaces and Continuity 8.3 Compactness 8.3.1 Closed Sets 8.3.2 Sequences and Limits 8.3.3 Interlude: Completeness 8.3.4 Compactness in Euclidean Space Uniform Continuity Distances from Sets 8.4 Quotients by Discrete Subgroups 8.5 Further Exercises 9 Connectedness, Smooth and Simple 9.1 Connectedness, Path and Simple 9.1.1 Homotopy; Simple Connectedness 9.2 Lifting Maps 9.2.1 The Winding Number 9.3 Differentiability: A Reminder 9.3.1 Mean Value Inequalities 9.3.2 Partial Derivatives 9.3.3 Inverse Functions 9.3.4 Second Derivatives 9.4 Differentiable Manifolds 9.5 Partitions of Unity 9.5.1 Proof of Theorem 9.66 9.6 Differentiable Connectedness 9.6.1 Piecewise Smooth Paths 9.7 Further Exercises 10 Path Integrals 10.1 Integrating Forms Along Paths Definition of the Integral Properties of the Integral 10.1.1 The Length of a Path 10.2 Integrating Along Smooth Paths 10.2.1 Linear Forms 10.2.2 Relating the General and Familiar Integrals The Fundamental Theorem of Calculus Derivative of an Integral Depending on a Parameter 10.3 Integrating Vector Fields 10.3.1 Conservative Vector Fields 10.3.2 The Winding Number Revisited 10.4 Symmetric Vector Fields 10.4.1 Missing a Point 10.5 Further Exercises 11 Complex Differentiation 11.1 Complex Derivatives and Integrals 11.1.1 Complex Integrals 11.2 Cauchy's Integral Formula Continuously Differentiable Functions and Primitives 11.2.1 Winding Numbers in the Complex Plane The Integral Formula 11.3 Uniform Convergence and Power Series 11.3.1 Absolute Convergence Rearrangements 11.3.2 Uniform Convergence Convergence on Compact Sets 11.3.3 Power Series 11.4 Analytic Functions 11.4.1 Differentiating Power Series 11.4.2 The Exponential and Trigonometric Functions 11.4.3 Continuously Differentiable Functions Are Analytic 11.5 Morera, Weierstrass, Liouville 11.5.1 Liouville's Theorem 11.6 Further Exercises 12 Riemann Surfaces 12.1 Holomorphic Surfaces 12.1.1 Meromorphic Functions The Meromorphic Conjugate 12.2 The Open Mapping Theorem 12.2.1 The Calculus of Residues 12.2.2 The Continuity of Roots of Polynomials 12.2.3 Open Mappings and Inverse Functions Consequences for Riemann Surfaces 12.3 Compact Riemann Surfaces 12.4 Riemann Surfaces for the Logarithm and Roots 12.4.1 The Logarithm 12.4.2 The Surface for the nth Root The Shift on 12.5 Analytic Continuation 12.6 Differential Forms on Surfaces 12.6.1 Pull-Backs of Meromorphic Forms 12.6.2 Quotients of Forms The Differential of a Meromorphic Function 12.6.3 Integration of Holomorphic Forms 12.7 Further Exercises Part III Curves and Surfaces 13 Curves Are Surfaces 13.1 The Implicit Function Theorem 13.2 Nonsingular Curves Are Riemann Surfaces 13.2.1 Vertical Parameterisations 13.2.2 An Atlas for the Nonsingular Part of a Curve 13.2.3 Rational Functions on Curves 13.2.4 Lifting Paths to Curves Algebraic Curves Are Connected 13.3 Intersections with Lines, Revisited 13.3.1 Continuous Intersection Multiplicities 13.3.2 Finding Intersection Points 13.3.3 Finding Intersecting Lines 13.3.4 An Application to Elliptic Curves 13.4 Further Exercises 14 Elliptic Functions and the Isomorphism Theorem 14.1 Elliptic Functions 14.1.1 The Weierstrass Function Definition of Is an Elliptic Function Inverse Images of Points 14.1.2 The Differential Equation for 14.2 The Curve E and the Isomorphism Theorem 14.2.1 The Isomorphism Theorem 14.3 Inversion 14.3.1 A Non-vanishing Form on a Nonsingular Cubic 14.3.2 Working After the Fact 14.3.3 Invariance of the Non-vanishing Holomorphic Form 14.3.4 Proof of the Inversion Theorem 14.4 Further Exercises 15 Puiseux Theory 15.1 Fractional Power Series and Their Holomorphic Functions 15.1.1 Formal and Informal Power Series Germs 15.1.2 Substitutions into Power Series 15.1.3 Fractional Power Series 15.1.4 The Holomorphic Function Defined by a Fractional Power Series The Induced Function on the Root Surface The Shift of a Fractional Power Series 15.2 Parameterisations of a Curve 15.2.1 n-Fold Parameterisations 15.2.2 Fractional Parameterisations 15.2.3 Existence of Parameterisations 15.3 Branches and Places 15.3.1 Central Places 15.3.2 Branches of a Curve 15.4 Puiseux Expansions and Factorisation into Places 15.4.1 Puiseux Expansions 15.4.2 The Implicit Definition of a Place A Factorisation of the Defining Polynomial 15.5 Intersection Multiplicities Using Places 15.5.1 Intersections of Curves and Places 15.5.2 Intersections of Curves 15.5.3 Orders and Tangents of Places 15.5.4 Some Nifty Consequences Intersections with Shifted Curves Intersection Multiplicity, Orders, and Shared Tangents 15.6 Further Exercises 16 A Brief History of Elliptic Functions 16.1 A History of Circles and Ellipses Interlude: The Circular Functions Inverting Elliptic Integrals Bibliography Index