# On Quaternionic Tori and their Moduli Spaces

## TL;DR

This paper classifies quaternionic tori, which are quotients of quaternions by lattices, by constructing a moduli space using slice regular functions and analyzing their automorphism groups.

## Contribution

It introduces a moduli space for quaternionic tori, classifies them via special lattice bases, and identifies automorphism groups and boundary subsets.

## Key findings

- Constructed a 12-dimensional moduli space for quaternionic tori.
- Identified all tori with non-trivial automorphism groups.
- Classified boundary points corresponding to automorphism groups.

## 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 \emph{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.