##
**Fall Semester, 2003**

**Course Title: Topics in Number Theory, Exponential Sums**
**Unique Number: M390C (57725)**
**Time and place: TTh 9:30-11:00 RLM 9.166
**
**Instructor:
Felipe Voloch
**

**Brief description:**
Exponential sums are sums of complex numbers of absolute value one.
They appear in many branches of Mathematics. We will be focusing on
Exponential sums in Number Theory and, in particular, on rational
exponential sums which are sums of roots of unity. We will discuss how
they are used and the basic questions about exponential sums, for instance,
estimates for their absolute values (both complex and p-adic).

Topics to be covered will include:

- Fourier analysis on finite abelian groups.
- Gauss sums and Jacobi sums. Hasse-Davenport formula. Evaluation
of Gauss sums. Stickelberger formula.
- Exponential sums, L-series and the Riemann hypothesis for curves.
- Incomplete exponential sums.
- p-adic properties of sums of roots of unity.

**Prerequisite: **
Graduate Algebra

**Textbook: **
None.
* *

**Course notes: **
Each student will be required to take notes for a week and TeX them.
The whole set of notes is now available
here

**Links**