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In this paper, we compare two cohomology groups associated to Shimuravarieties of PEL type, which are not proper over the base. One is the compactlysupported l-adic cohomology, and the other is the nearby cycle cohomology,namely, the compactly supported cohomology of the nearby cycle complex for thecanonical integral model of the Shimura variety over Z_p. We prove that theG(Q_p)-cuspidal part of these cohomology groups are the same, where G denotesthe reductive algebraic group naturally attached to the PEL datum. Someapplications to unitary Shimura varieties are also given.

Compactly supported cohomology and nearby cycle cohomology of open Shimura varieties of PEL type