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By studying modular invariance properties of some characteristic forms, which are related to elliptic genera, we obtain twisted cancellation formulas for characteristic forms. We apply these twisted cancellation formulas to study divisibilities on spin manifolds and congruences on spin c manifolds. In particular, we obtain twisted Rokhlin congruences for 8k + 4 dimensional spin c manifolds.

Modular invariance and twisted cancellations of characteristic numbers