  C. Voisin and C. Borcea have constructed mirror pairs of families of
Calabi-Yau threefolds by taking the quotient of the product of an elliptic
curve with a K3 surface endowed with a non-symplectic involution. In this
paper, we generalize the construction of Borcea and Voisin to any prime order
and build three and four dimensional Calabi-Yau orbifolds. We classify the
topological types that are obtained and show that, in dimension 4, orbifolds
built with an involution admit a crepant resolution and come in topological
mirror pairs. We show that for odd primes, there are generically no minimal
resolutions and the mirror pairing is lost.
