Integral representations of unramified Galois groups and matrix divisors over number fields
type: Item
instance of: scholarly article
title: Integral representations of unramified Galois groups and matrix divisors over number fields
zbMATH Open document ID: 0879.20023
author: Masanori Morishita
published in: Osaka Journal of Mathematics
publication date: 1997-11-30T00:00:00Z
review text: Let \(K\) be an algebraic number field with ring of integers \(\mathcal O\), \(\mathbf A\) its adele ring, \(U_n(\mathbf A)\) the maximal compact subgroup of \(GL_n(\mathbf A)\) and \(U_n(\mathcal O)=GL_n(K)\cap U_n(\mathbf A)\). Put \(G=\text{Gal}(\widetilde K/K)\), where \(\widetilde K\) is the maximal unramified extension of \(K\). In analogy to the ideas of \textit{A. Weil} [J. Math. Pures Appl., IX. Sér. 17, 47-87 (1938; Zbl 0018.06302)] for the study of the fundamental group of an algebraic curve, the author investigates the set \(R_n(\overline X)\) of integral unitary adelic representations \(\rho\colon G\to U_n(\mathcal O)\), where \(\overline X\) denotes the scheme \(\text{spec}(\mathcal O)\), completed at the infinite places. He constructs a map \(\overline{\Phi}_n\colon R_n(\overline X)\to Cl_n(\overline X)\) into the class group of \(n\)-dimensional matrix divisors, given by \(Cl_n(\overline X)=GL_n(K)\backslash GL_n(\mathbf A)/U_n(\mathbf A)\), and studies its properties. Finally, these ideas are demonstrated for the worked out example \(K=\mathbb{Q}(\sqrt{-155})\), in which case \(G\simeq S_3\).
reviewed by: Günter Lettl
Mathematics Subject Classification ID: 20G35
Mathematics Subject Classification ID: 11R32
Mathematics Subject Classification ID: 11R39
zbMATH DE Number: 871821
zbMATH Keywords: matrix divisors
zbMATH Keywords: integral unitary representations
zbMATH Keywords: adele rings
zbMATH Keywords: unramified extensions of number fields
zbMATH Keywords: maximal compact subgroups
zbMATH Keywords: class groups
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 871821