Abstract
Motivated by the theory of complex multiplication of abelian varieties, in this paper we study the conformality classes of flat tori in {mathbb {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}) 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.
Highlights
In this paper subgroup Γ we consider n-dimensional flat tori ∼= Zn induced by aTnΓ as lattice quotients Λ ⊂ Rn.of Rn under the action For this reason we of a use the notation TnΛ for the same torus
We will give examples of tori with conformal multiplications in higher dimensions, inspired by the fact that, in general, the corresponding conformal group of automorphisms is closely related to the symmetry group of the Voronoi region associated with the lattice that defines the torus
We study the class of conformally equivalent lattices and describe the moduli space of the corresponding tori and use a similar approach for biregular quaternionic tori to study their moduli space in higher dimension
Summary
In this paper subgroup Γ we consider n-dimensional flat tori ∼= Zn induced by a (maximal rank). We will give examples of tori with conformal multiplications in higher dimensions, inspired by the fact that, in general, the corresponding conformal group of automorphisms is closely related to the symmetry group of the Voronoi region associated with the lattice that defines the torus. This is due to the fact that if a finite group of SO(n) leaves invariant a lattice the group preserves the Voronoi region. All the examples of tori with conformal automorphisms are defined by lattices whose Voronoi regions (triangles, hexagons and squares) have many symmetries. We study the class of conformally equivalent lattices and describe the moduli space of the corresponding tori and use a similar approach (as in [4]) for biregular quaternionic tori to study their moduli space in higher dimension
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