**Credits** |
8 credit points |

**Instructors** |
Luijk, R.M. van (Universiteit Leiden), Stevenhagen, P. (Universiteit Leiden) |

**E-mail** |
rmluijk@gmail.com, psh@math.leidenuniv.nl |

**Aim** |
Along various historical paths, the origins of elliptic curves can be traced to calculus, complex analysis and algebraic geometry, and their arithmetic aspects have made them key objects in modern cryptography and in Wiles's proof of Fermat's last theorem. This course is an introduction to both the theoretical and the computational aspects of elliptic curves. |

**Description** |
The topics treated include a general discussion of elliptic curves and their group law, Diophantine equations in two variables, and Mordell's theorem. We will also discuss elliptic curves over finite fields with applications such as factoring integers, elliptic discrete logarithms, and cryptography. We will pursue both a theoretical and a computational approach. |

**Organization** |
Each class there will be three 45 minutes time slots: two lectures and one exercise class. |

**Examination** |
The final grade will be based on homework. |

**Literature** |
Literature (more details on the website, see Remarks) J.W.S. Cassels: Lectures on Elliptic Curves J.S. Milne: Elliptic Curves J.H. Silverman and J. Tate: Rational Points on Elliptic Curves J.H. Silverman: The arithmetic of elliptic curves J.H. Silverman: Advanced topics in the arithmetic of elliptic curves |

**Prerequisites** |
Linear algebra, groups, rings, fields. |

**Remarks** |
http://www.math.leidenuniv.nl/~rvl/elliptic/2011/. |