A higher-dimensional generalization of the Goss zeta function
type: Item
instance of: scholarly article
title: A higher-dimensional generalization of the Goss zeta function
zbMATH Open document ID: 0819.11055
published in: Journal of Number Theory
publication date: 1995-08-31T00:00:00Z
review text: This paper is both very original and very thought provoking. It contains the seeds of a quite remarkable generalization of the arithmetic of function fields of transcendency 1 over a finite field to function fields of arbitrary transcendency degree. It also poses very interesting challenges in non-archimedean analysis and arithmetic geometry as will be mentioned further on. We begin with a review of the transcendency 1 case: Let \(X\) be a smooth geometrically connected curve over the finite field \(\mathbb{F}_ q\) \((q=p^ t)\) with a fixed closed point \(\infty\). As is usual, we let \({\mathbf A}\) be the affine ring of \(X-\infty\), \({\mathbf k}\) the quotient field of \({\mathbf A}\) (= the function field of \(X\)) and \({\mathbf K}= {\mathbf k}_ \infty\) the completion of \({\mathbf k}\) at \(\infty\). Choose a fixed algebraic closure \(\overline {\mathbf K}\) of \({\mathbf K}\) with the canonical extension of \(| \cdot |_ \infty\) to it. Thus \({\mathbf K}\) is a local field and \({\mathbf A}\) is a discrete, co-compact subring of \({\mathbf K}\). The analogy -- in the 1-dimensional case -- is always with \(\mathbb{Z}\subset \mathbb{R}\). We let \(\pi\) be a fixed uniformizer in \({\mathbf K}\) (for instance, when \({\mathbf A}= \mathbb{F}_ q [T ]\) we may take \(\pi= 1/T\)). Let \(a\in {\mathbf k}^*\); one can then write \(a\) uniquely as \(a= \pi^ m u\) for some integer \(m\) and unit \(u\in {\mathbf K}^*\). We say \(a\) is ``positive'' (or ``monic'') if \(u\) is actually a 1-unit, in which case we set \(\langle a\rangle:= u\). Clearly \(\langle ab\rangle =\langle a\rangle \langle b\rangle\). Now set \(S_ \infty= \overline {\mathbf K}^* \times \mathbb{Z}_ p\); \(S_ \infty\) is a space of ``exponents'' in the same sense that \(\mathbb{C}\) is in classical number theory. Indeed, let \(a\) be positive with \(\deg a\) the degree of \(a\) \((= \deg_{\mathbb{F}_ q} {\mathbf A}/ (a))\) and let \(s= (x,y)\in S_ \infty\). Then we set \(a^ s:= x^{\deg a}\cdot \langle…
Mathematics Subject Classification ID: 11R58
Mathematics Subject Classification ID: 11G09
Mathematics Subject Classification ID: 11T55
Mathematics Subject Classification ID: 11G25
Mathematics Subject Classification ID: 14G15
zbMATH DE Number: 729087
zbMATH Keywords: exponentiation
zbMATH Keywords: Parshin completion
zbMATH Keywords: maximal flag
zbMATH Keywords: Goss zeta function
zbMATH Keywords: higher dimensional varieties
zbMATH Keywords: ample divisor
author: Mikhail Kapranov
reviewed by: David Goss
MaRDI profile type: Publication
full work available at URL: jnth.1995.1030
OpenAlex ID: W2060519341
DOI: 10.1006/JNTH.1995.1030
description: scientific article; zbMATH DE number 729087