AbstractMotivated by the theory of complex multiplication of abelian varieties, in this paper we study the conformality classes of flat tori in $${\mathbb {R}}^{n}$$
R
n
and investigate criteria to determine whether a n-dimensional flat torus has non trivial (i.e. bigger than $${\mathbb {Z}}^{*}={\mathbb {Z}}{\setminus }\{0\}$$
Z
∗
=
Z
\
{
0
}
) semigroup of conformal endomorphisms (the analogs of isogenies for abelian varieties). We then exhibit several geometric constructions of tori with this property and study the class of conformally equivalent lattices in order to describe the moduli space of the corresponding tori.
On the class-fields obtained by complex multiplication of abelian varieties
