We present an approach to proving control theorems for overconvergentautomorphic forms on some Harris-Taylor unitary Shimura varieties based on acomparison between the rigid coho- mology of the multiplicative ordinary locusand the rigid cohomology of the overlying Igusa tower, the latter which may becomputed using the Harris-Taylor version of the Langlands-Kottwitz method. Wealso prove a higher level version, generalizing work of Coleman.
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