## Elliptic CurvesAn elliptic curve is a particular kind of cubic equation in two variables whose projective solutions form a group. Modular forms are analytic functions in the upper half plane with certain transformation laws and growth properties. The two subjects--elliptic curves and modular forms--come together in Eichler-Shimura theory, which constructs elliptic curves out of modular forms of a special kind. The converse, that all rational elliptic curves arise this way, is called the Taniyama-Weil Conjecture and is known to imply Fermat's Last Theorem. Elliptic curves and the modeular forms in the Eichler- Shimura theory both have associated L functions, and it is a consequence of the theory that the two kinds of L functions match. The theory covered by Anthony Knapp in this book is, therefore, a window into a broad expanse of mathematics--including class field theory, arithmetic algebraic geometry, and group representations--in which the concidence of L functions relates analysis and algebra in the most fundamental ways. Developing, with many examples, the elementary theory of elliptic curves, the book goes on to the subject of modular forms and the first connections with elliptic curves. The last two chapters concern Eichler-Shimura theory, which establishes a much deeper relationship between the two subjects. No other book in print treats the basic theory of elliptic curves with only undergraduate mathematics, and no other explains Eichler-Shimura theory in such an accessible manner. |

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### Contents

Overview | 3 |

Curves in Projective Space | 19 |

Cubic Curves in Weierstrass Form | 50 |

Mordells Theorem | 80 |

Torsion Subgroup of EQ | 130 |

Complex Points 1 Overview | 151 |

Elliptic Functions | 152 |

Weierstrass p Function | 153 |

Geometry of the q Expansion | 227 |

Dimensions of Spaces of Modular Forms | 231 |

Function of a Cusp Form | 238 |

Petersson Inner Product | 241 |

Hecke Operators | 242 |

Interaction with Petersson Inner Product | 250 |

Modular Forms for Hecke Subgroups 1 Hecke Subgroups | 256 |

Modular and Cusp Forms | 261 |

Effect on Addition | 162 |

Overview of Inversion Problem | 165 |

Analytic Continuation | 166 |

Riemann Surface of the Integrand | 169 |

An Elliptic Integral | 174 |

Computability of the Correspondence | 183 |

Dirichlets Theorem 1 Motivation | 189 |

Dirichlet Series and Euler Products | 192 |

Fourier Analysis on Finite Abelian Groups | 199 |

Proof of Dirichlets Theorem | 201 |

Analytic Properties of Dirichlet L Functions | 207 |

Modular Forms for 512 Z 1 Overview | 221 |

Definitions and Examples | 222 |

Examples of Modular Forms | 265 |

Function of a Cusp Form | 267 |

Dimensions of Spaces of Cusp Forms | 271 |

Hecke Operators | 273 |

Oldforms and Newforms | 283 |

Function of an Elliptic Curve 1 Global Minimal Weierstrass Equations | 290 |

Zeta Functions and L Functions | 294 |

Hasses Theorem | 296 |

TaniyamaWeil Conjecture | 386 |

Notes | 401 |

409 | |

419 | |