Non-totally real number fields and toroidal groups

Abstract

In this paper we study the relationship between non-totally real number fields K and toroidal groups T, as well as meromorphic periodic functions, exploiting a representation of T as the generalized Jacobian ℑL(C) of a suitable elliptic curve C. We consider in detail the cubic and quartic cases. In these cases, we write down the relations between the minimal polynomial of a suitable primitive element of K and the parameters defining the generalized Jacobian ℑL(C) corresponding to the toroidal group associated with the ring of integers. Furthermore, for such a toroidal group we explicitly show the analytic and rational representations of its ring of endomorphisms, the former giving in turn a new (complex) representation of the ring of integers of K. Moreover, for the cubic case, we give an explicit description of the m-torsion of T in the geometric correspondence of T with ℑL(C), as image of a fractional ideal of K.