Demjanenko matrix, class number, and Hodge group
type: Item
instance of: scholarly article
title: Demjanenko matrix, class number, and Hodge group
zbMATH Open document ID: 0697.12003
DOI: 10.1016/0022-314X(90)90147-J
author: Fumio Hazama
published in: Journal of Number Theory
publication date: 1990-01-01T00:00:00Z
review text: For any odd prime number p, a modified Dem'yanenko matrix D is defined and shown to be equal to a matrix H, closely related to a representation matrix of the character group of the Hodge group of an abelian variety with complex multiplication by \({\mathbb{Q}}(\zeta_ p)\). The main result is the explicit expression: det D\(=\det H=(odd\) \(integer)\cdot h^-_ p\), where \(h^-_ p\) is the relative class number of \({\mathbb{Q}}(\zeta_ p)\), thus leading to the corollary that p is non-exceptional if \(h^-_ p\) is odd. This last result is of importance in the problem of giving an explicit bound for the order of torsion points on some elliptic curves.
Mathematics Subject Classification ID: 11R18
Mathematics Subject Classification ID: 11T22
Mathematics Subject Classification ID: 14H52
Mathematics Subject Classification ID: 11R23
Mathematics Subject Classification ID: 14K22
zbMATH DE Number: 4142151
zbMATH Keywords: Dem'yanenko matrix
zbMATH Keywords: Hodge group
zbMATH Keywords: abelian variety with complex multiplication
zbMATH Keywords: class number
zbMATH Keywords: order of torsion points
zbMATH Keywords: elliptic curves
reviewed by: W. W. J. Hulsbergen
MaRDI profile type: Publication
cites work: Maillet’s determinant
cites work: What is the rank of the Demjanenko matrix ?
cites work: Q3718827
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cites work: A note of Shimura's paper: Discontinuous groups and Abelian varieties
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cites work: On Maillet determinant
description: scientific article; zbMATH DE number 4142151