| |
| |
| Preface | |
| |
| |
| |
| First Ideas: Complex Manifolds, Riemann Surfaces, and Projective Curves | |
| |
| |
| |
| The Riemann Sphere | |
| |
| |
| |
| Complex Manifolds | |
| |
| |
| |
| Rational Functions | |
| |
| |
| |
| Luroth's Theorem | |
| |
| |
| |
| Automorphisms of P[superscript 1] | |
| |
| |
| |
| Spherical Geometry | |
| |
| |
| |
| Finite Subgroups and the Platonic Solids | |
| |
| |
| |
| Automorphisms of the Half-Plane | |
| |
| |
| |
| Hyperbolic Geometry | |
| |
| |
| |
| Projective Curves | |
| |
| |
| |
| Covering Surfaces | |
| |
| |
| |
| Scissors and Paste | |
| |
| |
| |
| Algebraic Functions | |
| |
| |
| |
| Examples | |
| |
| |
| |
| More on Uniformization | |
| |
| |
| |
| Compact Manifolds as Curves: Finale | |
| |
| |
| |
| Elliptic Integrals and Functions | |
| |
| |
| |
| Elliptic Integrals: Where They Come From | |
| |
| |
| |
| The Incomplete Integrals Reduced to Normal Form | |
| |
| |
| |
| The Complete Integrals: Landen, Gauss, and the Arithmetic-Geometric Mean | |
| |
| |
| |
| The Complete Elliptic Integrals: Legendre's Relation | |
| |
| |
| |
| The Discovery of Gauss and Abel | |
| |
| |
| |
| Periods in General | |
| |
| |
| |
| Elliptic Functions in General | |
| |
| |
| |
| The and-Function | |
| |
| |
| |
| Elliptic Integrals, Complete and Incomplete | |
| |
| |
| |
| Two Mechanical Applications | |
| |
| |
| |
| The Projective Cubic | |
| |
| |
| |
| The Problem of Inversion | |
| |
| |
| |
| The Function Field | |
| |
| |
| |
| Addition on the Cubic | |
| |
| |
| |
| Abel's Theorem | |
| |
| |
| |
| Jacobian Functions: Reprise | |
| |
| |
| |
| Covering Tori | |
| |
| |
| |
| Finale: Higher Genus | |
| |
| |
| |
| Theta Functions | |
| |
| |
| |
| Jacobi's Theta Functions | |
| |
| |
| |
| Some Identities | |
| |
| |
| |
| The Jacobi and Weierstrass Connections | |
| |
| |
| |
| Projective Embedding of Tori | |
| |
| |
| |
| Products | |
| |
| |
| |
| Sums of Two Squares | |
| |
| |
| |
| Sums of Four Squares | |
| |
| |
| |
| Euler's Identities: Partitio Numerorum | |
| |
| |
| |
| Jacobi's and Higher Substitutions | |
| |
| |
| |
| Quadratic Reciprocity | |
| |
| |
| |
| Ramanujan's Continued Fractions | |
| |
| |
| |
| Modular Groups and Modular Functions | |
| |
| |
| |
| The Modular Group of First Level | |
| |
| |
| |
| The Modular Group of Second Level | |
| |
| |
| |
| Fundamental Cells | |
| |
| |
| |
| Generating the Groups | |
| |
| |
| |
| Gauss on Quadratic Forms | |
| |
| |
| |
| The Group of Anharmonic Ratios | |
| |
| |
| |
| Modular Forms | |
| |
| |
| |
| Eisenstein Sums | |
| |
| |
| |
| Absolute Invariants | |
| |
| |
| |
| Triangle Functions | |
| |
| |
| |
| The Modular Equation of Level 2 | |
| |
| |
| |
| Landen's Transformation | |
| |
| |
| |
| Modular Equations of Higher Level | |
| |
| |
| |
| Jacobi's Modular Equation | |
| |
| |
| |
| Jacobi and Legendre's Derivation: Level 5 | |
| |
| |
| |
| Arithmetic Subgroups: Overview | |
| |
| |
| |
| Ikosaeder and the Quintic | |
| |
| |
| |
| Solvability of Equations of Degree [less than] 4 | |
| |
| |
| |
| Galois Groups Revisited | |
| |
| |
| |
| The Galois Group of Level 5 | |
| |
| |
| |
| An Element of Degree 5 | |
| |
| |
| |
| Hermite on the Depressed Equation | |
| |
| |
| |
| Hermite on the Quintic | |
| |
| |
| |
| A Geometric View | |
| |
| |
| |
| Imaginary Quadratic Number Fields | |
| |
| |
| |
| Algebraic Numbers | |
| |
| |
| |
| Primes and Ideal Numbers | |
| |
| |
| |
| Class Invariants and Kronecker's Jugendtraum | |
| |
| |
| |
| Application of the Modular Equation | |
| |
| |
| |
| The Class Polynomial | |
| |
| |
| |
| Class Invariants at a Prime Level | |
| |
| |
| |
| Irreducibility of the Class Polynomial | |
| |
| |
| |
| Class Field and Galois Group | |
| |
| |
| |
| Computation of the Class Invariants | |
| |
| |
| |
| Arithmetic of Elliptic Curves | |
| |
| |
| |
| Arithmetic of the Projective Line | |
| |
| |
| |
| Cubics: The Mordell--Weil Theorem | |
| |
| |
| |
| Examples | |
| |
| |
| |
| Proof of the Mordell--Weil Theorem | |
| |
| |
| References | |
| |
| |
| Index | |