We fix a prime number l. Let F be an imaginary CM number field and F + is the total real subfield. We assume that every prime of F + over p splits in F. Let OF + be the ring of integers of F +. We fix a definite unitary group G over OF +. Follwing David Geraghty’s thesis, I will talk about the construction of spaces of automorphic forms over G, ordinary Hecke algebras and related Hida control theorems.

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