Demjanenko matrix, class number, and Hodge group
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.
abelian variety with complex multiplication
scientific article; zbMATH DE number 4142151