Vanderbilt University

Date | Topics covered | Notes |
---|---|---|

August 24 | Introduction | Lecture 1 Notes |

August 26 | Motivating Exampls of Modular Forms | Lecture 1 Notes |

August 28 | Elliptic Functions: Definitions and basic facts | Lecture 2 Notes |

August 31 | Elliptic Functions: Constructions and the Weierstrass p-function | Lecture 3 Notes |

September 2 | The field of elliptic functions and a special differential equation | Lecture 4 Notes |

September 4 | Elliptic Curves | Lecture 5 Notes |

September 7 | More on Elliptic Curves | Lecture 6 Notes |

September 9 | Connections with Ellitpic Functions and the Congruent Number Problem | Lecture 7 Notes |

September 11 | Torsion points on the congruent number elliptic curves | Lecture 8 Notes |

September 14 | Modular Forms: The basics | Lecture 9 Notes |

September 16 | The fundamental domain | Lecture 10 Notes |

September 18 | Eisenstein Series | Lecture 11 Notes |

September 21 | The Valence Formula | Lecture 12 Notes |

September 23 | Dimensions of modular form spaces | Lecture 12 Notes |

September 25 | Modular functions and elliptic curves | Lecture 13 Notes |

September 28 | Differential operators and quasimodular forms | Lecture 14 Notes |

September 30 | The Ramanujan tau function | Lecture 15 Notes |

October 2 | The Dedekind eta function | Lecture 16 Notes |

October 5 | Partition functions and Tauberian theorems | Lecture 17 Notes |

October 7 | Hecke operators | Lecture 18 Notes |

October 9 | New modualr forms from old ones | Lecture 18 Notes |

October 12 | More on Hecke operators | Lecture 19 Notes |

October 14 | Bases of eigenforms and the Petersson inner product | Lecture 20 Notes |

October 16 | L-functions and their analytic continuation and functional equations | Lecture 21 Notes |

October 19 | Theta functions, Poisson summation | Lecture 22 Notes |

October 21 | Sums of square identities, generalized theta functions | Lecture 22 Notes |

October 23 | Generalized theta functions | Lecture 22 Notes |

October 26 | Weil Conjectures and the zeta function of the congruent number curves | Lecture 23 Notes |

October 28 | Gauss and Jacobi sums and the zeta function of the congruent number curves | Lecture 23 Notes |

October 30 | L-functions of elliptic curves, and connections with modular forms (Modularity Theorem/Birch and Swinnerton-Dyer Conjecture) | Lecture 24 Notes |

November 2 | Atkin-Lehner-Li Theory, Eichler-Shimura theory, and motivating ideas behind the proof of Fermat's Last Theorem | Lecture 25 Notes |

November 4 | Motivating ideas behind the proof of Fermat's Last Theorem | Lecture 25 Notes |

November 6 | Applications to the congruent number problem | Lecture 26 Notes |

November 9 | Class Cancelled | |

November 11 | Poincaré series of different types | Lecture 27 Notes |

November 13 | The Shimura/Shintani correspondence and Tunnell's Theorem on congruent numbers | Lecture 27 Notes |

November 16 | Complex Multiplication | Lecture 28 Notes |

November 18 | Singular Moduli | Lecture 28 Notes |

November 20 | Class polynomials | Lecture 28 Notes |

November 30 | Class number relations and Kronecker's Jugendtraum | Lecture 28 Notes |

December 2 | Moonshine | Lecture 29 Notes |

December 4 | Sphere packing | Lecture 30 Notes |