On the ideal class group problem for global fields
type: Item
instance of: scholarly article
title: On the ideal class group problem for global fields
zbMATH Open document ID: 0933.11053
author: Marc Perret
published in: Journal of Number Theory
publication date: 2000-03-27T00:00:00Z
review text: The ideal class group problem is the classical question: given a finite abelian group \(G\), does there exist a number field \(K\) having \(G\) as the ideal class group of \(K\)? This is an open problem. \textit{G. Cornell} [Number theory, Proc. Conf., Carbondale 1979, Lect. Notes Math. 751, 82-88 (1979; Zbl 0423.12009)] proved that given \(G\) a finite abelian group, there exists a cyclotomic field \(K_1\) such that \(G\) is a quotient of the ideal class group of \(K_1\). Furthermore, \textit{O. Yahagi} [Tokyo J. Math. 1, 275-283 (1978; Zbl 0403.12011)] proved that if \(G\) is an \(\ell\)-group, there exists a field \(K_2\) such that \(G\) is the \(\ell\)-Sylow subgroup of the ideal group of \(K_2\). For function fields, \textit{H. Stichtenoth} [Arch. Math. 32, 336-340 (1979; Zbl 0404.12011)] proved that if \(G\) is an abelian group of exponent \(n\) and cardinality \(\geq n ^{2 (48n/e) ^4}\), then there does not exist any smooth projective irreducible algebraic curve \(X\), defined over any finite field having \(G\) as its divisor class group of degree zero. In this paper the author considers the analogous problem for \(S\)-class groups of number fields and also for geometric ideal class groups of congruence function fields. He proves a reduction lemma that states that if \(K\) is a global field, \(S\) is a finite set of places (containing the archimedian places) and \(L\) is an intermediate field \(K \subset L \subset K _S ^{\text{Hilb}}\), then there exists a finite set \(S'\) of places of \(K\) such that \(L = K _{S\cup S'}^{\text{Hilb}}\). Using this lemma and Cornell's theorem in the number field case and results of \textit{B. Anglès} [J. Number Theory 70, 146-159 (1998; Zbl 0911.11052)] in the function field case, the author proves that the problem has an affirmative answer in both cases.
Mathematics Subject Classification ID: 11R29
Mathematics Subject Classification ID: 11R58
Mathematics Subject Classification ID: 11R65
zbMATH DE Number: 1309264
zbMATH Keywords: class groups
zbMATH Keywords: ideal class groups
zbMATH Keywords: \(S\)-class groups
zbMATH Keywords: geometric ideal class groups
zbMATH Keywords: congruence function fields
reviewed by: Gabriel D. Villa Salvador
MaRDI profile type: Publication
OpenAlex ID: W2018516000
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DOI: 10.1006/JNTH.1998.2366
description: scientific article; zbMATH DE number 1309264