Algebraic varieties over finite fields are considered from the point of view of their invariants such as the number of points of a variety that are defined over the ground field and its extensions. The case of curves has been actively studied over the last thirty-five years, and hundreds of papers have been devoted to the subject. In dimension two or higher, the situation becomes much more difficult and has been little explored. This survey presents the main approaches to the problem and describes a major part of the known results in this direction. Bibliography: 102 titles.
- algebraic varieties over finite fields
- arithmetic statistics
- error-correcting codes
- explicit formulae in arithmetic
- points on surfaces
- zeta functions