  The first purpose of this paper is to examine the relationship between
equivariant elliptic genera and orbifold elliptic genera. We apply the
character theory of Hopkins et. al. to the Borel-equivariant genus associated
to the sigma orientation of Ando-Hopkins-Strickland to define an orbifold genus
for certain total quotient orbifolds and supersingular elliptic curves. We show
that our orbifold genus is given by the same sort of formula as the orbifold
``two-variable'' genus of Dijkgraaf et al. In the case of a finite cyclic
orbifold group, we use the characteristic series for the two-variable genus to
define an analytic equivariant genus in Grojnowski's equivariant elliptic
cohomology, and we show that this gives precisely the orbifold two-variable
genus. The second purpose of this paper is to study the effect of varying the
BU<6>-structure in the Borel-equivariant sigma orientation. We show that
varying the BU<6>-structure by a class in the third cohomology of the orbifold
group produces discrete torsion in the sense of Vafa. This result was first
obtained by Sharpe, for a different orbifold genus and using different methods.
