Caliber of a global field
type: Item
instance of: scholarly article
title: Caliber of a global field
zbMATH Open document ID: 0666.12010
DOI: 10.1016/0022-314X(88)90002-9
author: Roger Paysant-Le Roux
published in: Journal of Number Theory
publication date: 1988-01-01T00:00:00Z
review text: \textit{G. Lachaud} introduced in the paper ``Calibre et fonctions zeta des corps quadratiques réels'' [Prépublication Nice, No. 63 (1984)] the notion of caliber of a quadratic number field \(K\). The caliber, \(m(J)\) of an ideal class \(J\) is the length of the period of the continued fraction expansion of \(w_ 1/w_ 2\) where \(J=\mathbb Z[w_ 1,w_ 2]\). The caliber of \(K\), \(m(K)\), is the sum of the \(m(J)\) over all ideal classes. In the paper this notion of caliber is extended to any global field (number field or function field over a finite field of constants). The definition is technical, it is based on the notions of extremal points and reduced modules. It is shown that there are only finitely many number fields of given degree and caliber. This result is based on the following theorem of Siegel: \(\log (hR)\sim \log (D_ K^{1/2}),\) for \(D_ K\to \infty\). Here \(h\) is the class number, \(R\) the regulator and \(D_ K\) the discriminant of \(K\). In the paper the following inequality is shown: \[ hR\leq C[m(K)]^{s-1} \] where \(C=(s-1)^{(s-1)/2}(\log ((2^ t/\pi D_ K^{1/2}))^{s-1}\), \(s=\) number of archimedean places and \(2t=\) number of complex imbeddings of \(K\). So by Siegel's theorem it follows that if \(m(K)\) is bounded, so is \(D_ K\). This implies the required finiteness result.
reviewed by: Jan Van Geel
Mathematics Subject Classification ID: 11R27
Mathematics Subject Classification ID: 11R29
Mathematics Subject Classification ID: 11H99
zbMATH DE Number: 4089673
zbMATH Keywords: caliber of a quadratic number field
zbMATH Keywords: ideal class
zbMATH Keywords: discriminant
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4089673