Caliber of a global field
\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.
scientific article; zbMATH DE number 4089673