On the ring of endomorphisms of a certain class of quasi-Abelian varieties arising from non-totally real number fields

In this paper we study the relationship between isogeny classes of toroidal groups and non-totally real number fields. Under some assumptions concerning the discriminant, we prove that any number field with one pair of complex embeddings arises as the ring of endomorphisms (tensorized with [Formula: see text]) of a toroidal group of complex and real ranks differing by one. On the other hand, any toroidal group [Formula: see text], with [Formula: see text], such that the field [Formula: see text] is of the aforementioned type, arises from [Formula: see text]. In particular, a representation of the toroidal group [Formula: see text] arising from the field [Formula: see text] is provided through a period matrix in standard coordinates of [Formula: see text], expressed in terms of the roots and the coefficients of (a multiple of) the minimal polynomial of a suitable primitive element of [Formula: see text]. Furthermore, for such a toroidal group we present the analytic and rational representations of its ring of endomorphisms. The same problem is also studied in the more intricate case of a field having many pairs of complex embeddings. We consider in detail the quintic case.