Generalizing the theorem of Nakagawa on binary cubic forms to number fields

Abstract

The goal of this thesis is to study possible generalizations of a theorem of Nakagawa, first stated as a conjecture by Ohno, that gives a relationship between cubic fields of positive and negative discriminant. This theorem is described as an equation of Dirichlet series whose coefficients are class numbers of binary cubic forms. Its proof makes an extensive use of class field theory. Our approach for generalizing this result to cubic extensions of an arbitrary number field is to write the series in terms of ideles following the works of Datskovsky and Wright. By comparing the residues at their poles, we are able to deduce a conjecture that is a direct generalization of the original theorem. In the process of refining this generalization, we obtain some results concerning local integrals and series over idele group character. Moreover, we use tables of number fields which are currently available and computer algebra systems to provide strong evidence for the validity of the proposed conjecture.