Sieve methods for varieties over finite fields and arithmetic schemes

Abstract

Classical sieve methods of analytic number theory have recently been adapted to a geometric setting. In the new setting, the primes are replaced by the closed points of a variety over a finite field or more generally of a scheme of finite type over Z. We will present the method and some of the surprising results that have been proved using it. For instance, the probability that a plane curve over F2 is smooth is asymptotically 21/64 as its degree tends to infinity. Much of this paper is an exposition of results in [Poo04] and [Ngu05].