Arithmetic of Weighted Diagonal Surfaces over Finite Fields

Abstract

In this paper, we discuss several fundamental properties of weighted diagonal surfaces over finite fields. Weighted diagonal surfaces are defined to be diagonal surfaces in weighted projective 3-spaces. In general, they have cyclic quotient singularities. For our arithmetic investigations, we choose their minimal resolutions. We are particularly interested in their Picard numbers and the orders of their Brauer groups. First we prove that the (congruence) zeta-functions of minimal resolutions of weighted diagonal surfaces can be described in terms of twisted Jacobi sums. The proof is based on an explicit description of the set of rational points in weighted projective spaces. Secondly, we give a formula for the Picard numbers of minimal resolutions of weighted diagonal surfaces. The Tate conjecture asserts that their Picard numbers are equal to the orders of certain poles of their zeta-functions. Thus, we first prove the Tate conjecture for the minimal resolutions. Then we calculate the orders of poles of their zeta-functions. Here the validity of the conjecture will be established, in fact, for more general surfaces. Thirdly, we look into quantities associated to special values of zeta-functions, especially the orders of Brauer groups, of certain minimal resolutions of weighted diagonal surfaces. Specifically, we confine ourselves to the minimal resolutions whose Néron–Severi groups have Z-bases consisting only of canonical divisors and exceptional divisors. For such surfaces, we compute the orders of Brauer groups in terms of twisted Jacobi sums.