On quaternionic tori and their moduli space

Quaternionic tori are defined as quotients of the skew field H of quaternions by rank-4 lattices. Using slice regular functions, these tori are endowed with natural structures of quaternionic manifolds (in fact quaternionic curves), and a moduli space in a 12-dimensional real space is then constructed to classify them up to biregular diffeomorphisms. The points of the moduli space correspond to suitable special bases of rank-4 lattices, which are studied with respect to the action of the group GL(4,Z), and up to biregular diffeomeorphisms. All tori with a non trivial group of biregular automorphisms and all possible groups of their biregular automorphisms are then identified, and recognized to correspond to five different subsets of boundary points of the moduli space.