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Demjanenko matrix, class number, and Hodge group. (English) Zbl 0697.12003
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.
Reviewer: W.Hulsbergen

MSC:
11R18 Cyclotomic extensions
11T22 Cyclotomy
14H52 Elliptic curves
11R23 Iwasawa theory
14K22 Complex multiplication and abelian varieties
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