We present an approach to proving control theorems for overconvergent automorphic forms on Harris-Taylor unitary Shimura varieties based on a comparison between the rigid cohomology of the multiplicative ordinary locus and the rigid cohomology of the overlying Igusa tower, the latter which may be computed using the Harris-Taylor version of the Langlands-Kottwitz method. We also prove a higher level version, generalizing work of Coleman.
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