Algebraic curves and Riemann surfaces for undergraduates : the theory of the donut

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 The Theory of the Donut, in a Nutshell 1.2 Overview of the Book Part I: Algebraic Curves Affine and Projective Curves Intersections of Curves Elliptic Curves Part II: Riemann Surfaces Three Kinds of Surfaces Analytic Functions Real Analysis, Complex Analysis, and Path Integrals Finally, Riemann Surfaces Part III: Curves and Surfaces 1.3 Preliminaries, and Some Notation Part I Algebraic Curves 2 Algebra 2.1 Polynomials and Power Series The Category of Rings Simplifying Notation Derived Properties Back to Formal Power Series Infinite Sums Several Variables More on Polynomials The Degree of a Polynomial Polynomial Substitution 2.2 Unique Factorisation Divisibility in Integral Domains Divisibility in R[x] Unique Factorisation Domains Multisets Unique Factorisation Unique Factorisation in Polynomial Rings One Variable Interlude: Algebraically Closed Fields Gauss's Lemmas 2.3 Groups The Category of Groups Subgroups Group Homomorphisms Quotient Groups Cyclic Groups The Characteristic of a Ring The Symmetric Group 2.4 Linear Algebra Over Integral Domains Matrices, Linear Spaces, and Linear Maps Linear Spaces Linear Maps Invertible and Nonsingular Matrices Dimension and Complements The Determinant The Effect of Row Operations Polynomial Substitution Detecting Singularity 2.5 Further Exercises 3 Affine Space 3.1 Definition of Hypersurfaces 3.2 The Resultant The Sylvester Matrix The Resultant, Common Roots, and More Variables Adding More Variables The Resultant is a Linear Combination 3.3 Study's Lemma 3.4 Affine Lines and Rational Parameterisations Affine Lines Rational Parameterisations 3.5 Further Exercises 4 Projective Space 4.1 Homogeneous Polynomials 4.2 Projective Space 4.3 Projective Lines and Maps Projective Maps 4.4 Embedding Affine Space into Projective Space 4.5 Changes of Coordinates Change of Variable Four Point Lemma 4.6 Spaces of Curves The Dual Plane Principle of Duality 4.7 Products of Projective Spaces 4.8 Further Exercises 5 Tangents 5.1 Introduction: Affine Tangents and Intersections with Lines Intersection Multiplicities Homogeneous Coordinates 5.2 Formal Partial Derivatives Properties of Derivatives The Chain Rule Euler's Relation Taylor Expansions The Discriminant 5.3 Higher Order Tangents The Moduli Space of Tangents Invariance of the Higher Order Tangent 5.4 The Intersection of a Line with a Curve Definition of Intersection Multiplicity Bézout for a Line Invariance of Multiplicity of Intersection with a Line Affine Calculations Tangents and Intersections with Lines Defining Multiplicities Using Tangents Simple Intersections Are the Norm 5.5 Further Exercises 6 Bézout's Theorem 6.1 A First Look at the Intersection of Curves The Resultant of Homogeneous Polynomials Is Homogeneous A Weak Version of Bézout's Theorem A Naïve Definition of Intersection Multiplicity 6.2 The Homogeneous Resultant Main Property of the Homogeneous Resultant 6.3 Multiplicity of Intersection and Bézout's Theorem Coding Lines in P2P2 The Resultant of the General Intersection Polynomials Bihomogeneity of Rf,g The Structure of the Hypersurface Defined by Rf,g Intersection Multiplicity and Bézout's Theorem The Intersection Multiset Geometric Invariance 6.4 Coincidence with Earlier Definitions Using the Family of Vertical Lines Intersecting Lines 6.5 Categoricity of Multiplicity of Intersection 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 Flexes and the Second Order Tangent The Hessian 7.2 The Group Operation on a Nonsingular Cubic Curve The Complement Curve Associativity of the Group Operation 7.3 Normal Forms for Nonsingular Cubics Explicit Calculations of the Group Operation 7.4 Further Exercises Part II Riemann Surfaces 8 Quasi-Euclidean Spaces 8.1 Topology of Rn 8.2 Manifolds Topology of Pre-manifolds Subspaces The Hausdorff Property Topological Countability Manifolds Spaces and Continuity 8.3 Compactness Closed Sets Sequences and Limits Interlude: Completeness Compactness in Euclidean Space Uniform Continuity Distances from Sets 8.4 Quotients by Discrete Subgroups Discrete Subgroups of Rn Quotients by Discrete Subgroups The Torus Topological Groups 8.5 Further Exercises 9 Connectedness, Smooth and Simple 9.1 Connectedness, Path and Simple Homotopy; Simple Connectedness 9.2 Lifting Maps The Winding Number 9.3 Differentiability: A Reminder Mean Value Inequalities Partial Derivatives Inverse Functions Second Derivatives 9.4 Differentiable Manifolds 9.5 Partitions of Unity 9.6 Differentiable Connectedness Piecewise Smooth Paths 9.7 Further Exercises 10 Path Integrals 10.1 Integrating Forms Along Paths Definition of the Integral Properties of the Integral Concatenation of Paths The Length of a Path 10.2 Integrating Along Smooth Paths 10.3 Integrating Vector Fields Conservative Vector Fields The Winding Number Revisited 10.4 Symmetric Vector Fields Missing a Point 10.5 Further Exercises 11 Complex Differentiation 11.1 Complex Derivatives and Integrals Complex Integrals 11.2 Cauchy's Integral Formula Winding Numbers in the Complex Plane The Integral Formula 11.3 Uniform Convergence and Power Series Absolute Convergence Rearrangements Uniform Convergence Convergence on Compact Sets Power Series 11.4 Analytic Functions Differentiating Power Series The Exponential and Trigonometric Functions Continuously Differentiable Functions Are Analytic 11.5 Morera, Weierstrass, Liouville 11.6 Further Exercises 12 Riemann Surfaces 12.1 Holomorphic Surfaces Meromorphic Functions The Meromorphic Conjugate 12.2 The Open Mapping Theorem The Calculus of Residues The Continuity of Roots of Polynomials Open Mappings and Inverse Functions Consequences for Riemann Surfaces 12.3 Compact Riemann Surfaces 12.4 Riemann Surfaces for the Logarithm and Roots The Logarithm The Surface for the nth Root The Shift on 12.5 Analytic Continuation 12.6 Differential Forms on Surfaces 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 Vertical Parameterisations An Atlas for the Nonsingular Part of a Curve Rational Functions on Curves Lifting Paths to Curves Algebraic Curves Are Connected 13.3 Intersections with Lines, Revisited Continuous Intersection Multiplicities Finding Intersection Points Finding Intersecting Lines An Application to Elliptic Curves 13.4 Further Exercises 14 Elliptic Functions and the Isomorphism Theorem 14.1 Elliptic Functions The Weierstrass Function Definition of Is an Elliptic Function Inverse Images of Points The Differential Equation for 14.2 The Curve E and the Isomorphism Theorem The Isomorphism Theorem 14.3 Inversion A Non-vanishing Form on a Nonsingular Cubic Working After the Fact Invariance of the Non-vanishing Holomorphic Form Proof of the Inversion Theorem 14.4 Further Exercises 15 Puiseux Theory 15.1 Fractional Power Series and Their Holomorphic Functions Formal and Informal Power Series Germs Substitutions into Power Series Fractional Power Series 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 n-Fold Parameterisations Fractional Parameterisations Existence of Parameterisations 15.3 Branches and Places Central Places Branches of a Curve 15.4 Puiseux Expansions and Factorisation into Places Puiseux Expansions The Implicit Definition of a Place A Factorisation of the Defining Polynomial Remark on Newton Polygons 15.5 Intersection Multiplicities Using Places Intersections of Curves and Places Intersections of Curves Orders and Tangents of Places Some Nifty Consequences Intersections with Shifted Curves Intersection Multiplicity, Orders, and Shared Tangents 15.6 Further Exercises 16 A Brief History of Elliptic Functions Bibliography Index