Modular invariance, characteristic numbers and $\eta$ invariants

We generalize the “miraculous cancellation” formulas of Alvarez-Gaum'e, Witten and Kefeng Liu to a twisted version where an extra complex line bundle is involved. We also apply our result to discuss intrinsic relations between the higher dimensional Rokhlin type congruence formulas of Ochanine [O1], Finashin [F] and Zhang [Z1, 2]. In particular, an analytic proof of the Finashin congruence formula is given.