  Let $M$ be a $Spin$-manifold with $S^1$-action and let $\sigma \in S^1$ be of
finite order. We show that the indices of certain twisted Dirac operators
vanish if the action of $\sigma $ has sufficiently large fixed point
codimension. These indices occur in the Fourier expansion of the elliptic genus
of $M$ in one of its cusps. As a by-product we obtain a new proof of a theorem
of Hirzebruch and Slodowy on involutions.
