Another look at the index formulas of cyclotomic number theory
type: Item
instance of: scholarly article
title: Another look at the index formulas of cyclotomic number theory
zbMATH Open document ID: 0872.11047
author: Greg W. Anderson
published in: Journal of Number Theory
publication date: 1997-09-09T00:00:00Z
review text: The author considers a full cyclotomic field \(K={\mathbb{Q}}(e^{2\pi i/f})\) and introduces a module \(\mathcal L\) over \({\mathbb{Z}}[G]\), where \(G\) is the Galois group of \(K\) over \(\mathbb{Q}\). This module combines in a beautiful way the information afforded by cyclotomic units and the Stickelberger ideal. More concisely: There is an explicit canonical module \(\mathcal M\) containing \(\mathcal L\); the index \([{\mathcal M}:{\mathcal L}]\) equals the class number \(h_K\) (let me call this result ``Anderson's index formula''); and one can extract both the ``Stickelberger quotient'' and the quotient ``units modulo cyclotomic units'' from the pair of modules \(\mathcal M\) and \(\mathcal L\). The author proves his index formula by means of Dedekind's residue formula and a very elegant calculation of this residue which avoids using \(L\)-functions explicitly. The link with universal distributions, and with Sinnott's index formulas for \(h^+\) and \(h^-\), is lucidly explained. It should be mentioned that the ``correcting'' powers of 2 in Sinnott's index formulas arise from cohomology which comes in automatically when one takes plus parts and minus parts, and in this respect Anderson's index formula is simpler, not containing any correction factors at all. The idea behind the construction of \(\mathcal L\) is the following: \(\mathcal L\) is the span of all \(l(r):=-\log \bigl(1-1\otimes e^{2\pi ir/f}\bigr)\) with \(1\leq r \leq f-1\). (Here one works in \(V={\mathbb{R}}\otimes K\), and log\((1-z)\) is defined by the standard series \(-\sum_{n\geq1}z^n/n\).) From this one already ``sees'' the cyclotomic units. The Stickelberger ideal comes in at the back door: If we map \(V\) to \({\mathbb{C}}G\) via \(1\otimes x \mapsto \sum_{(a,f)=1}\sigma_a(x)\cdot \sigma_a^{-1}\), then \(l(r)-l(-r)\) maps to \(2\pi i\) times a Stickelberger element (slightly adjusted). This is stated in the paper as formula 7 (without comment), and it does become clear if one thinks abou…
reviewed by: Cornelius Greither
Mathematics Subject Classification ID: 11R33
Mathematics Subject Classification ID: 11R18
Mathematics Subject Classification ID: 11R29
zbMATH DE Number: 934973
zbMATH Keywords: cyclotomic units
zbMATH Keywords: Stickelberger ideal
zbMATH Keywords: index formulas
zbMATH Keywords: distributions
MaRDI profile type: Publication
full work available at URL: jnth.1996.0118
OpenAlex ID: W1984635755
DOI: 10.1006/JNTH.1996.0118
description: scientific article; zbMATH DE number 934973