| Preface | p. xi |
| First Ideas: Complex Manifolds, Riemann Surfaces, and Projective Curves | p. 1 |
| The Riemann Sphere | p. 1 |
| Complex Manifolds | p. 3 |
| Rational Functions | p. 7 |
| Luroth's Theorem | p. 8 |
| Automorphisms of P[superscript 1] | p. 12 |
| Spherical Geometry | p. 14 |
| Finite Subgroups and the Platonic Solids | p. 16 |
| Automorphisms of the Half-Plane | p. 24 |
| Hyperbolic Geometry | p. 25 |
| Projective Curves | p. 27 |
| Covering Surfaces | p. 30 |
| Scissors and Paste | p. 33 |
| Algebraic Functions | p. 41 |
| Examples | p. 46 |
| More on Uniformization | p. 51 |
| Compact Manifolds as Curves: Finale | p. 52 |
| Elliptic Integrals and Functions | p. 54 |
| Elliptic Integrals: Where They Come From | p. 55 |
| The Incomplete Integrals Reduced to Normal Form | p. 62 |
| The Complete Integrals: Landen, Gauss, and the Arithmetic-Geometric Mean | p. 65 |
| The Complete Elliptic Integrals: Legendre's Relation | p. 68 |
| The Discovery of Gauss and Abel | p. 71 |
| Periods in General | p. 77 |
| Elliptic Functions in General | p. 81 |
| The and-Function | p. 84 |
| Elliptic Integrals, Complete and Incomplete | p. 87 |
| Two Mechanical Applications | p. 89 |
| The Projective Cubic | p. 92 |
| The Problem of Inversion | p. 93 |
| The Function Field | p. 95 |
| Addition on the Cubic | p. 98 |
| Abel's Theorem | p. 104 |
| Jacobian Functions: Reprise | p. 109 |
| Covering Tori | p. 113 |
| Finale: Higher Genus | p. 118 |
| Theta Functions | p. 125 |
| Jacobi's Theta Functions | p. 125 |
| Some Identities | p. 127 |
| The Jacobi and Weierstrass Connections | p. 131 |
| Projective Embedding of Tori | p. 133 |
| Products | p. 135 |
| Sums of Two Squares | p. 140 |
| Sums of Four Squares | p. 142 |
| Euler's Identities: Partitio Numerorum | p. 143 |
| Jacobi's and Higher Substitutions | p. 147 |
| Quadratic Reciprocity | p. 150 |
| Ramanujan's Continued Fractions | p. 154 |
| Modular Groups and Modular Functions | p. 159 |
| The Modular Group of First Level | p. 159 |
| The Modular Group of Second Level | p. 160 |
| Fundamental Cells | p. 162 |
| Generating the Groups | p. 166 |
| Gauss on Quadratic Forms | p. 167 |
| The Group of Anharmonic Ratios | p. 169 |
| Modular Forms | p. 172 |
| Eisenstein Sums | p. 176 |
| Absolute Invariants | p. 177 |
| Triangle Functions | p. 183 |
| The Modular Equation of Level 2 | p. 185 |
| Landen's Transformation | p. 187 |
| Modular Equations of Higher Level | p. 189 |
| Jacobi's Modular Equation | p. 192 |
| Jacobi and Legendre's Derivation: Level 5 | p. 198 |
| Arithmetic Subgroups: Overview | p. 200 |
| Ikosaeder and the Quintic | p. 206 |
| Solvability of Equations of Degree [less than] 4 | p. 206 |
| Galois Groups Revisited | p. 207 |
| The Galois Group of Level 5 | p. 209 |
| An Element of Degree 5 | p. 212 |
| Hermite on the Depressed Equation | p. 214 |
| Hermite on the Quintic | p. 216 |
| A Geometric View | p. 217 |
| Imaginary Quadratic Number Fields | p. 224 |
| Algebraic Numbers | p. 225 |
| Primes and Ideal Numbers | p. 227 |
| Class Invariants and Kronecker's Jugendtraum | p. 235 |
| Application of the Modular Equation | p. 237 |
| The Class Polynomial | p. 239 |
| Class Invariants at a Prime Level | p. 243 |
| Irreducibility of the Class Polynomial | p. 248 |
| Class Field and Galois Group | p. 249 |
| Computation of the Class Invariants | p. 250 |
| Arithmetic of Elliptic Curves | p. 252 |
| Arithmetic of the Projective Line | p. 252 |
| Cubics: The Mordell--Weil Theorem | p. 253 |
| Examples | p. 255 |
| Proof of the Mordell--Weil Theorem | p. 259 |
| References | p. 265 |
| Index | p. 278 |
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