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
Time permitting, we will have an introduction to the statement of the
Weil conjectures and the results of Dwork, Grothendieck and Deligne.
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.