Kempf.04.03

Abstract: ``Counting a number'' is the basic of mathematics. In this lecture, the following type of counting is discussed: given a Diophantine equation defined by a set of polymonials defined over a finite field, we want to count the number of solutions in it. This basic question in arithmetic is related to another counting problem in topology: counting the number of fixed points of a continuous map, a problem treated by ``Lefschetz trace formula'' in terms of cohomology theory. The connection was found by A. Weil about sixty years ago. I will explain the background, then focus on this trace formula approach in abstract algebraic geometry (Grothendieck-Verdier theory), and review the developments of Lefschetz-Verdier trace formula by now. In particular I will focus on a formula conjectured by P. Deligne in 1980's, which is extremely useful in many areas of Mathematics. I will also try to explain briefly a hidden geometric viewpoint behind it -rigid analytic geometry, an analytic theory for abstract algebraic varieties.