Anderson's module for cyclotomic fields of prime conductor
type: Item
instance of: scholarly article
title: Anderson's module for cyclotomic fields of prime conductor
zbMATH Open document ID: 0888.11045
author: Pietro Cornacchia
published in: Journal of Number Theory
publication date: 1998-05-27T00:00:00Z
review text: In 1996, \textit{G. Anderson} [J. Number Theory 60, 142-164 (1996; Zbl 0872.11047)] introduced a module which combines in a nice way two modules which are important in the theory of abelian fields: the so-called Stickelberger quotient \(S\), and the quotient \(E/C\) of units by cyclotomic units. The former quotient is intimately related to the minus class group of the corresponding abelian field, and the latter to the plus class group. Away from the two-primary part, Anderson's module simply decomposes as a direct sum of the two parts just described, but in the two-primary part, the extension which arises is in general nontrivial, and new questions arise. The author of the paper under review considers the case of a cyclotomic field \(L\) of prime conductor \(l\) and studies its Anderson module \(\mathcal A\). From the above general discussion one is led to conclude that \(\mathcal A\) should be closely related to the whole class group Cl\((L)\). Anderson proved that these two abelian groups have the same order. The author shows a nice result pointing in this precise direction: the modules \(\mathcal A\) and Cl\((L)\) have the same Tate cohomology over \(G\) (the Galois group of \(L\)). Then it is shown that the Pontryagin dual \({\mathcal A}^*\) of the finite module \(\mathcal A\) is cyclic over \(Z[G]\), so \({\mathcal A}^*\) is isomorphic to \(Z[G]/J\) for a unique ideal \(J\). More intrinsically, \(J\) can be gotten as the (first) Fitting ideal of \(\mathcal A\). (\(\mathcal A\) itself need not be cyclic. Similar statements concerning \(E/C\) can be made.) He puts forward a conjecture that \(J\) annihilates Cl\((L)\), and the even stronger conjecture that \(J\) is the Fitting ideal of Cl\((L)\). The latter seems to be a difficult problem, although there is a lot of support for it, in the reviewer's opinion. The author proves that 2\(J\) annihilates Cl\((L)\). An interesting report on explicit calculations concludes the paper.
Mathematics Subject Classification ID: 11R33
Mathematics Subject Classification ID: 11R29
Mathematics Subject Classification ID: 11R18
zbMATH DE Number: 1109651
zbMATH Keywords: class groups
zbMATH Keywords: units
zbMATH Keywords: Stickelberger ideal
zbMATH Keywords: Galois cohomology
zbMATH Keywords: Stickelberger quotient
zbMATH Keywords: cyclotomic units
zbMATH Keywords: minus class group
zbMATH Keywords: plus class group
zbMATH Keywords: Anderson's module
zbMATH Keywords: cyclotomic field
zbMATH Keywords: Tate cohomology
reviewed by: Cornelius Greither
MaRDI profile type: Publication
full work available at URL: jnth.1997.2184
OpenAlex ID: W2007074462
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DOI: 10.1006/JNTH.1997.2184
description: scientific article; zbMATH DE number 1109651