  The theory of coverings of the two-dimensional torus is a standard part of
algebraic topology and has applications in several topics in string theory, for
example, in topological strings. This paper initiates applications of this
theory to the counting of orbifolds of toric Calabi-Yau singularities, with
particular attention to Abelian orbifolds of C^D. By doing so, the work
introduces a novel analytical method for counting Abelian orbifolds, verifying
previous algorithm results. One identifies a p-fold cover of the torus T^{D-1}
with an Abelian orbifold of the form C^D/Z_p, for any dimension D and a prime
number p. The counting problem leads to polynomial equations modulo p for a
given Abelian subgroup of S_D, the group of discrete symmetries of the toric
diagram for C^D. The roots of the polynomial equations correspond to orbifolds
of the form C^D/Z_p, which are invariant under the corresponding subgroup of
S_Ds. In turn, invariance under this subgroup implies a discrete symmetry for
the corresponding quiver gauge theory, as is clearly seen by its brane tiling
formulation.
