Rank one elliptic \(A\)-modules and \(A\)-harmonic series
type: Item
instance of: scholarly article
title: Rank one elliptic \(A\)-modules and \(A\)-harmonic series
zbMATH Open document ID: 0807.11032
DOI: 10.1215/S0012-7094-94-07321-3
author: Greg W. Anderson
published in: Duke Mathematical Journal
publication date: 1995-02-26T00:00:00Z
review text: This paper is a tour de force on the part of the author. It represents a high point of a trend started by Drinfeld: Apply ideas from the theory of partial differential equations to the arithmetic of function fields over finite fields; in particular, basic to the author's work is the concept of a ``soliton'' -- an idea from the theory of water waves and the famous nonlinear Korteweg-de Vries equation (KdV): \[ {{\partial\varphi} \over {\partial t}} = {{\partial^ 3\varphi} \over {\partial x^ 3}} +6\varphi{{\partial\varphi} \over {\partial x}}. \] (A ``soliton'' is -- roughly -- a periodic or quasi-periodic solution to KdV.) Let \(X/\mathbb{F}_ q\) be a smooth, projective, geometrically irreducible curve and let \(\infty\in X\) be a rational point; one then sets \(A\) to be the affine ring of \(X-\infty\) and \(k\) to be the quotient field of \(A\). By fiat the field \(k\) contains exactly one place not in \(\text{Spec} (A)\) thus mimicking the classical case of \(\mathbb{Z}\subset \mathbb{Q}\). The field \(k\) forms a ``bottom'' for the theory, exactly as \(\mathbb{Q}\) is a bottom in algebraic number theory. In this situation we can define characteristic \(p\) analogs of classical \(L\)-functions and \(\Gamma\)-functions [see the reviewer in `The arithmetic of function fields', de Gruyter, 313-402 (1992)]. In order to make these definitions ``good'' analytic functions (in a straightforward sense) one always ``groups according to degree''. For instance, an \(L\)-series \(\sum c_ a a^{-s}\) is analytically continued by grouping the terms of the same degree; \[ \sum c_ a a^{- s}= \sum_ j \Bigl( \sum_{\deg a=j} c_ a a^{-s} \Bigr), \] etc. Thus each such function \(f\) is made up of a sequence \(f_ k\) of ``degree parts''. The author seeks to study such \(f\) via an ``interpolation formula'': One seeks an algebraic function \(\varphi\) of two variables \(\{x,t\}\) such that \(f_ k\) (or, perhaps, \(f_{k+1}/ f_ k\) or \(f_{k+2} f_ k/ f^ 2_{k+1}\)) is recovered by specializ…
Mathematics Subject Classification ID: 11G09
Mathematics Subject Classification ID: 35Q51
Mathematics Subject Classification ID: 35Q53
zbMATH DE Number: 572042
zbMATH Keywords: Drinfeld modules
zbMATH Keywords: soliton
zbMATH Keywords: arithmetic of function fields
zbMATH Keywords: Korteweg-de Vries equation
zbMATH Keywords: regulators for zeta-functions
reviewed by: David Goss
MaRDI profile type: Publication
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full work available at URL: s0012-7094-94-07321-3
OpenAlex ID: W1501125363
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description: scientific article; zbMATH DE number 572042