##
**Spring Semester, 2001**

**Course Title: Topics in Algebra, Equations over
finite fields**
**Unique Number: M390C (55320)**
**Time and place: TTh 12:30-2:00 RLM 10.176
**
**Instructor:
Felipe Voloch
**

**Brief description:**
We will study the classical topic of counting or estimating the number
of solutions to (systems of) polynomial equations over finite fields.
We will first review the basic theory of finite fields and study some
elementary and combinatorial bounds, such as the Chevalley-Warning
theorem and generalizations.

We will then move to the theory of curves over finite fields, prove
Weil's analogue of the Riemann hypothesis and discuss some improvements
to it.
Time permitting, we will have an introduction to the statement of the
Weil conjectures and the results of Dwork, Grothendieck and Deligne.

### Class notes taken by students

The only requirement for this class, for registered students, is to
take turns to write up notes in TeX that I will make available to all
other students through this webpage. The notes are not meant to be in
final form and I would encourage you to send
me any corrections or comments.
These notes have now been put in a uniform shape (by Brian van der Ven)
and edited to remove some glaring errors and misprints (by me). You can
download the whole thing in one file
pdf.

**Prerequisite: **
graduate algebra

**Textbook: **none
* *

### Links

- Milne's
notes on Étale cohomology.
- Katz's
notes on the Weil conjectures.
- Tables
of curves with many points.
- Oesterlé's bound, a Pari script,
courtesy of K. Lauter.