# SCHEDULE

# | Date | Topic (references) | Materials |
---|---|---|---|

1 | 1/31 | Introduction | notes, slides |

2 | 2/2 | The group law, Weierstrass and Edwards equations (Washington 2.1-3, 2.6.3, Bernstein-Lange 2007, Bernstein-Lange 2010) | notes, worksheet 1, worksheet 2 |

3 | 2/7 | Finite field arithmetic (Modern Computer Algebra 3.2, 8.1-4, 9.1, 11.1, 14.2-6, HEHCC 9.1-2, Rabin 1980, Cantor-Zassenhaus 1981) | notes, slides, worksheet, bonus video |

4 | 2/9 | Isogenies (Washington 2.9, Silverman III.4) | notes, slides |

5 | 2/14 | Isogeny kernels and division polynomials (Washington 3.2, 12.3, Silverman III.4) | notes, slides, worksheet |

6 | 2/16 | Endomorphism rings (Washington 4.2, , Silverman III.6) | notes, slides |

7 | 2/22 | Hasse's Theorem, point counting (Washington 4.3) | notes, slides |

8 | 2/23 | Schoof's algorithm (Washington 4.2, 4.5, Schoof 1985) | notes, slides, worksheet |

9 | 2/28 | Generic algorithms for the discrete logarithm problem (Washington 5.2, Pohlig-Hellman 1978, Pollard 1978, Shor 1994, Shoup 1997) | notes, slides |

10 | 3/2 | Index calculus, smooth numbers, factoring integers (Washington 5.1, 7.1, Granville, Lenstra) | notes, slides, worksheet 1, worksheet 2, worksheet 3 |

11 | 3/7 | Elliptic curve primality proving (ECPP) (Washington 7.2, Goldwasser-Kilian, Pomerance) | notes, slides |

12 | 3/9 | Endomorphism algebras (Silverman III.9) | notes, slides |

13 | 3/14 | Ordinary and supersingular curves (Silverman III.1,V, Washington 2.7, 4.6) | notes, slides |

14 | 3/16 | Elliptic curves over C (part I) (Cox Sec. 10, Silverman VI.2-3, Washington 9.1-2) | notes, slides |

15 | 3/28 | Elliptic curves over C (part II) (Cox Sec. 10-11, Silverman VI.4-5, Washington 9.2-3) | notes, slides |

16 | 3/30 | Complex multiplication (CM) (Cox Sec. 11, Silverman VI.5, Washington 9.3) | notes |

17 | 4/4 | The CM torsor (Cox Sec. 7, Silverman (advanced topics) II.1.1) | notes |

18 | 4/11 | Riemann surfaces and modular curves (Silverman (advanced topics) I.2, Milne V.1) | slides, notes |

19 | 4/13 | The modular equation (Cox Sec. 11, Milne V.2, Washington pp. 273-274) | slides, notes |

20 | 4/20 | The Hilbert class polynomial (Cox Sec. 8, 11) | slides, notes |

21 | 4/25 | Ring class fields and the CM method (Cox 8, 11) | slides, notes |

22 | 4/25 | Isogeny volcanoes (Sutherland) | slides, notes |

23 | 5/2 | The Weil pairing (Miller, Washington 11, Silverman III.8) | slides, notes |

24 | 5/4 | Modular forms and L-functions (Milne V.3-4) | slides, notes |

25 | 5/9 | Fermat's last theorem (Milne V.7-9, Washington 15, Cornell-Stevens-Silverman I) | slides, notes |