Zeta functions of algebraic cycles over finite fields

Abstract

In this paper, we generalize the concept of a zeta function from zero cycles to higher dimensional cycles and investigate its meromorphic continuation and order of pole at t = 1 by means of a Riemann-Roch approach. We begin with an example which motivated the present work. A classical problem about finite fields is to find a formula for the number N~(d, n) of irreducible polynomials over Fq of degree d in n variables as d varies. This question goes back to Gauss (see the notes of Chapter 3 in [21]), who first found a formula in the case when there is only one variable and the finite field is a prime field Fp. Dickson generalized Gauss's formula from a prime finite field to a general finite field Fq. Since then, at tempts have been made to find a formula in the case of several variables. In 1936, Carlitz ([2]) found an analogous formula for the number of irreducible factorizable polynomials of degree d, where a polynomial over Fq is factorizable if it factors as a product of linear factors over the algebraic closure of Fq. This generalizes Gauss's formula in some sense. Later, Carlitz ([3-4]) established the asymptotic behavior of g'(d,n) (as d --, c~). Some of Carlitz's results were generalized by Cohen ([6-7I). Both Carlitz [3-4] and Cohen [6-7] said that no explicit formula for N~(d, n) is known in the several variable case. Using ideas from the theory of zeta functions, we found an explicit but complicated combinatorial formula for Nr n). The proof actually shows that in the several variable case, there is no combinatorial formula of the Gauss type; rather, there is a nice p-adic formula generalizing Gauss's formula. In some sense, this answers the question of Carlitz and Cohen. To understand this, we consider homogeneous polynomials. Let Nq (d, n) be the number of homogeneous irreducible polynomials over Fq of degree d in n