On $p$-Adic Zeta-Functions Associated to the Positive Topology of Algebraic Number Fields

Abstract

In [S] we introduced a new Grothendieck topology (called positive topology) over the rings of integers of algebraic number fields. If K is a finite extension of Q then Spec(0^) furnished with the positive topology shows a behavior very similar to the etale site over a complete curve over a finite field.In this paper we are going to investigate /7-adic zeta functions associated to the positive topology. Following the analogy it is natural to expect a functional equation. The aim of this paper is to show that such a functional equation holds true as soon as a primitive/?*' 1 root of unity is contained in K.If K/k is an abelian extension then the positive /?-adic zeta-function of K splits into a product of positive p-adic L-functions attached to the characters of K/k in the usual way. If the field k is totally real and "totally /7-real" (e.g. k = Q, see below for the definition) we calculate (at least up to the Greenberg conjecture) the positive j?-adic L-functions L p p os (s,\l/) in terms of the analytic /7-adic L-functions defined by Kubota/Leopoldt and Deligne/Ribet. §1. Review of the Positive TopologyIn this section we want to summarize the basic definitions and results about the positive topology over the rings of integers of algebraic number fields. Details and proofs can be found in [S]. Let p be a prime number. We