The zeta function of a cubic surface over a finite field

Abstract

Introduction. The object of this paper is to obtain an explicit formula for the zeta function of an arbitrary non-singular cubic surface over a finite field. Let k denote the finite field of q elements, and kn the field of qn elements which is the unique algebraic extension of k of degree n. Let be a non-singular variety defined over k, and for each n > 0 let be the number of points defined over kn which lie on . The zeta function of is given byDwork has shown in (3) that for any this is a rational function of q−s; and in particular it follows from the results he proves in (4) that if is a non-singular cubic surface thenand hence alsoHere the numbers w o depend only on q and on .