| List of Figures | |
| List of Tables | |
| Preface | |
| Standard Notation | |
| Overview | p. 3 |
| Curves in Projective Space | |
| Projective Space | p. 19 |
| Curves and Tangents | p. 24 |
| Flexes | p. 32 |
| Application to Cubics | p. 40 |
| Bezout's Theorem and Resultants | p. 44 |
| Cubic Curves in Weierstrass Form | |
| Examples | p. 50 |
| Weierstrass Form, Discriminant, j-invariant | p. 56 |
| Group Law | p. 67 |
| Computations with the Group Law | p. 74 |
| Singular Points | p. 77 |
| Mordell's Theorem | |
| Descent | p. 80 |
| Condition for Divisibility by 2 | p. 85 |
| E(Q)/2E(Q), Special Case | p. 88 |
| E(Q)/2E(Q), General Case | p. 92 |
| Height and Mordell's Theorem | p. 95 |
| Geometric Formula for Rank | p. 102 |
| Upper Bound on the Rank | p. 107 |
| Construction of Points in E(Q) | p. 115 |
| Appendix on Algebraic Number Theory | p. 122 |
| Torsion Subgroup of E(Q) | |
| Overview | p. 130 |
| Reduction Modulo p | p. 134 |
| p-adic Filtration | p. 137 |
| Lutz-Nagell Theorem | p. 144 |
| Construction of Curves with Prescribed Torsion | p. 145 |
| Torsion Groups for Special Curves | p. 148 |
| Complex Points | |
| Overview | p. 151 |
| Elliptic Functions | p. 152 |
| Weierstrass p Function | p. 153 |
| Effect on Addition | p. 162 |
| Overview of Inversion Problem | p. 165 |
| Analytic Continuation | p. 166 |
| Riemann Surface of the Integrand | p. 169 |
| An Elliptic Integral | p. 174 |
| Computability of the Correspondence | p. 183 |
| Dirichlet's Theorem | |
| Motivation | p. 189 |
| Dirichlet Series and Euler Products | p. 192 |
| Fourier Analysis on Finite Abelian Groups | p. 199 |
| Proof of Dirichlet's Theorem | p. 201 |
| Analytic Properties of Dirichlet L Functions | p. 207 |
| Modular Forms for SL(2,Z) | |
| Overview | p. 221 |
| Definitions and Examples | p. 222 |
| Geometry of the q Expansion | p. 227 |
| Dimensions of Spaces of Modular Forms | p. 231 |
| L Function of a Cusp Form | p. 238 |
| Petersson Inner Product | p. 241 |
| Hecke Operators | p. 242 |
| Interaction with Petersson Inner Product | p. 250 |
| Modular Forms for Hecke Subgroups | |
| Hecke Subgroups | p. 256 |
| Modular and Cusp Forms | p. 261 |
| Examples of Modular Forms | p. 265 |
| L Function of a Cusp Form | p. 267 |
| Dimensions of Spaces of Cusp Forms | p. 271 |
| Hecke Operators | p. 273 |
| Oldforms and Newforms | p. 283 |
| L Function of an Elliptic Curve | |
| Global Minimal Weierstrass Equations | p. 290 |
| Zeta Functions and L Functions | p. 294 |
| Hasse's Theorem | p. 296 |
| Eichler-Shimura Theory | |
| Overview | p. 302 |
| Riemann surface X[subscript 0](N) | p. 311 |
| Meromorphic Differentials | p. 312 |
| Properties of Compact Riemann Surfaces | p. 316 |
| Hecke Operators on Integral Homology | p. 320 |
| Modular Function j([tau]) | p. 333 |
| Varieties and Curves | p. 341 |
| Canonical Model of X[subscript 0](N) | p. 349 |
| Abstract Elliptic Curves and Isogenies | p. 359 |
| Abelian Varieties and Jacobian Variety | p. 367 |
| Elliptic Curves Constructed from S[subscript 2]([Gamma](N)) | p. 374 |
| Match of L Functions | p. 383 |
| Taniyama-Weil Conjecture | |
| Relationships among Conjectures | p. 386 |
| Strong Weil Curves and Twists | p. 392 |
| Computations of Equations of Weil Curves | p. 394 |
| Connection with Fermat's Last Theorem | p. 397 |
| Notes | p. 401 |
| References | p. 409 |
| Index of Notation | p. 419 |
| Index | p. 423 |
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