- Lecturer
- Prof. Dr. Emmanuel Kowalski
- Organizer
- Dante Bonolis
- Lectures
- Tue 08-10 - HG E 22 , Fri 10-12 - HG G 26.5
- Exercise session
- Every second Friday starting from 21.09.2018

Exponential sums over finite fields arise in many problems of number theory. We will discuss the elementary aspects of the theory (centered on the Riemann Hypothesis for curves, following Stepanov's method) and survey the formalism arising from Deligne's general form of the Riemann Hypothesis over finite fields. We will then discuss various applications, especially in analytic number theory. The goal is to understand both the basic results on exponential sums in one variable, and the general formalism of Deligne and Katz that underlies estimates for much more general types of exponential sums, including the "trace functions" over finite fields.

- Examples of elementary exponential sums
- The Riemann Hypothesis for curves and its applications
- Definition of trace functions over finite fields
- The formalism of the Riemann Hypothesis of Deligne
- Selected applications

We prepare an exercise sheet every two weeks. The new exercises will be posted here every other Thursday. We expect you to look at the problems and to prepare questions for the exercise class one week after. Please hand in your solutions in the exercise class.

exercise sheet | due by | solutions | |
---|---|---|---|

Exercise sheet 1 | October 5 | Solutions 1 | |

Exercise sheet 2 | October 19 | Solutions 2 | |

Exercise sheet 3 | October 30 | Chapter 6 in Exponential sums over finite fields, I: elementary methods | |

Exercise sheet 4 | November 16 | Solutions 4 | |

Exercise sheet 5 | November 30 | Solutions 5 | |

Exercise sheet 6 | December 21 | Solutions 6 | |

The first exercise class will be in the first week of the semester, Friday 21th of September.

time | room | assistant | language |
---|---|---|---|

Fr 10-12 | HG G 26.5 | Dante Bonolis | English |

- Emmanuel Kowalski: Exponential sums over finite fields, I: elementary methods, https://people.math.ethz.ch/~kowalski/exp-sums.pdf
- Étienne Fouvry, Emmanuel Kowalski, Philippe Michel: Trace functions over finite fields and their applications, https://people.math.ethz.ch/~kowalski/trace-functions-pisa.pdf
- Emmanuel Kowalski: Trying to understand Deligne’s proof of the Weil conjectures, https://people.math.ethz.ch/~kowalski/deligne.pdf
- Enrico Bombieri: Counting points on curves over finite fields, http://www.numdam.org/item/SB_1972-1973__15__234_0