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

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.