# Exponential Sums over Finite Fields Autumn 2018

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.

### Short Outline

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