Modular Invariance and Twisted Anomaly Cancellations of Characteristic   Numbers
  By studying modular invariance properties of some characteristic forms, we obtain twisted anomaly cancellation formulas. We apply these twisted cancellation formulas to study divisibilities on spin manifolds and congruences on spin$^c$ manifolds. Especially, we get twisted Rokhlin congruences for $8k+4$ dimensional spin$^c$ manifolds 
Qingtao Chen, Fei Han