Analytic Number Theory -- Trace Functions over Finite Fields

First lecture: Wednesday, September 17, 2014 during the exercise class.

Lecture notes: http://www.math.ethz.ch/~kowalski/trace-functions.html

Change of Schedule

Please note the following changes:

There are no lectures on Monday 22.09.2014 and 29.09.2014.

There is a regular exercise class on Wednesday 24.09.2014.

There is a lecture on Wednesday 1.10.2014 instead of the exercise class.

Contents

Trace functions over finite fields, and sums of trace functions, appear in many contexts of analytic number theory, from the study of primes to automorphic forms and L-functions. They contain both classical exponential sums over finite fields, families of these, and other functions of "algebraic nature". Their study is deeply linked with the Riemann Hypothesis over finite fields.

The goal is to understand the underlying theory of trace functions over finite fields, as well as the way they are used in a range of applications to analytic number theory.

The course will cover the following topics:

(1) Introduction and motivation: some examples of trace functions and where they appear

(2) The formalism of trace functions over finite fields

(3) The Riemann Hypothesis over finite fields

(4) Applications to analytic number theory

Prerequisites

Algebra I, Mass und Integral; Commutative algebra and/or algebraic geometry would be useful.

Exercises

First exercise session: 24.09.2014

Literature

• The papers of Fouvry, Kowalski and Michel
  
• P. Deligne, "Sommes trigonométriques"
  
• N. Katz, "Gauss sums, Kloosterman sums, and monodromy groups"
  
• N. Katz, "Exponential sums and differential equations"
  
• N. Katz, "Sommes exponentielles"Joseph H. Silverman and John Tate, "Rational Points on Elliptic Curves"