On quaternionic tori and their moduli space

Abstract

Quaternionic tori are defined as quotients of the skew field $\mathbb{H}$ 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 fundamental region in a 12-dimensional real subspace 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$(4,\mathbb{Z})$, 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.