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 Q) of a toroidal group of complex and real ranks differing by one. On the other hand, any toroidal group T, with rk_R T= rk_C T+1, such that the field K:=End_0(T) is of the aforementioned type, arises from K. In particular, a representation of the toroidal group T arising from the field K is provided through a period matrix in standard coordinates of T, expressed in terms of the roots and the coefficients of (a multiple of) the minimal polynomial of a suitable primitive element of K. 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.