Log-algebraicity of twisted \(A\)-harmonic series and special values of \(L\)-series in characteristic \(p\)
type: Item
instance of: scholarly article
title: Log-algebraicity of twisted \(A\)-harmonic series and special values of \(L\)-series in characteristic \(p\)
zbMATH Open document ID: 0868.11031
author: Greg W. Anderson
published in: Journal of Number Theory
publication date: 1997-08-19T00:00:00Z
review text: Let \(R\) be an arbitrary commutative ring with unit \(1_R\). There is an obvious map from \(\mathbb{Z}\to R\) which has \(1\mapsto 1_R\); thus a dual morphism of affine schemes \(\text{Spec}(R)\to \text{Spec}(\mathbb{Z})\). Therefore every scheme may thought of as lying over \(\text{Spec}(\mathbb{Z})\). In practice this means that ``\(\mathbb{Z}\)-concepts'' such as abelian group, class group, unit groups, etc., can be translated to arbitrary schemes. In other words, classical arithmetic geometry is based on concepts derived from \(\mathbb{Z}\), which is not terribly surprising. Now let \(\mathcal X\) be a projective, smooth, geometrically connected curve over the finite field \(\mathbb{F}_q\) \((q=p^n)\). Let \(\infty\in{\mathcal X}\) be a closed point and \({\mathbf A}\) the affine ring of those functions on \({\mathcal X}\) regular outside \(\infty\); let \({\mathbf k}\) be the function field of \({\mathcal X}\) (= fraction field of \({\mathbf A}\)). In this situation one has the theory of Drinfeld \({\mathbf A}\)-modules where \({\mathbf A}\) plays the role of \(\mathbb{Z}\), \({\mathbf k}\) of \(\mathbb{Q}\), and so on. By fiat, \({\mathbf A}\) is the ``bottom'' of the theory; that is, the theory is defined only for schemes lying over \(\text{Spec} ({\mathbf A})\). The importance of the paper being reviewed is that, for the first time, objects related to ``\({\mathbf A}\)-concepts'' are created for Drinfeld modules over global \({\mathbf A}\)-fields. More precisely, the paper presents the technology needed to find, in certain basic circumstances, the \({\mathbf A}\)-analog of ``units'' and ``cyclotomic units'' for rank one Drinfeld modules. It therefore opens up many fascinating areas of inquiry and some of them, such as an amazing analog of the classical Kummer-Vandiver conjecture, will be mentioned below. As is known to beginning calculus students, one has \(\exp(- \sum^\infty_{n=1} z^n/n)=1-z\). In number theory this, and its \(p\)-adic analogue, are used t…
Mathematics Subject Classification ID: 11G09
Mathematics Subject Classification ID: 11R58
Mathematics Subject Classification ID: 11R27
Mathematics Subject Classification ID: 11R70
zbMATH DE Number: 934974
zbMATH Keywords: log-algebraicity of twisted \(A\)-harmonic series
zbMATH Keywords: special values of \(L\)-series in characteristic \(p\)
zbMATH Keywords: exponential map
zbMATH Keywords: root numbers
zbMATH Keywords: special points for Drinfeld modules
zbMATH Keywords: function field
zbMATH Keywords: rank one Drinfeld modules
zbMATH Keywords: Kummer-Vandiver conjecture
zbMATH Keywords: Carlitz module
zbMATH Keywords: Carlitz module analog of cyclotomic units
zbMATH Keywords: unit group
reviewed by: David Goss
MaRDI profile type: Publication
full work available at URL: jnth.1996.0119
OpenAlex ID: W2139210940
DOI: 10.1006/JNTH.1996.0119
description: scientific article; zbMATH DE number 934974