We present an approach to proving control theorems for overconvergent
automorphic forms on some Harris-Taylor unitary Shimura varieties based on a
comparison between the rigid coho- mology 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.Comment: 25 pages. Main results strengthened, higher level version include