  We show that the Atiyah-Patodi-Singer reduced $\eta$-invariant of the twisted
Dirac operator on a closed $4m-1$ dimensional spin manifold, with the twisted
bundle being the Witten bundle appearing in the theory of elliptic genus, is a
meromorphic modular form of weight $2m$ up to an integral $q$-series. We prove
this result by combining our construction of certain modular characteristic
forms associated to a generalized Witten bundle on spin$^c$-manifolds with a
deep topological theorem due to Hopkins.
