## Javier Fresán: Catalogue data in Autumn Semester 2016 |

Name | Prof. Dr. Javier Fresán (Professor Sorbonne Université - Paris) |

Address | École Polytechnique CMLS 91128 Palaiseau FRANCE |

Department | Mathematics |

Relationship | Lecturer |

Number | Title | ECTS | Hours | Lecturers | |
---|---|---|---|---|---|

401-3570-66L | Algebraic Number Theory Number of participants limited to 12. | 4 credits | 2S | J. Fresán | |

Abstract | Much of the progress in algebraic number theory aimed at solving concrete Diophantine equations. At the heart of the problem lies the fact that the ring of integers of a number field does not have unique factorization. The "class group" measures how much this property fails. The seminar will present basic results around this invariant, including finiteness and the relation to zeta functions. | ||||

Learning objective | |||||

Content | The following topics will be covered: - The quadratic reciprocity law - The geometry of numbers - Integral quadratic forms - Number fields and rings of integers - Finiteness of the class number - Unique factorization of ideals - The Dedekind zeta function of a number field and the class number formula The seminar will be (probably) followed by a more advanced course on Class Field Theory on the Spring Semester. | ||||

Literature | Our basic reference will be chapters I and VII of Neukirch's book "Algebraic number theory" (Grundlehren Math. Wiss. 322. Springer-Verlag, Berlin, 1999). Additional references will be given at the beginning of the seminar. | ||||

Prerequisites / Notice | Basic knowledge of algebraic structures (groups, rings, fields) and Galois theory, at the level of Algebra I and II. More advanced topics will be explained when needed. | ||||

401-4145-66L | Reading Course: Abelian Varieties over Finite Fields | 2 credits | 4A | J. Fresán, P. S. Jossen | |

Abstract | |||||

Learning objective |