Spherical functions on p-adic homogeneous spaces and those applications

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We intend to investigated certain$ p$-adic homogeneous spaces together with spherical functions and have number theoretic applications. We have approached this purpose in the following way : 1) to formulate expression formulas of spherical functions which can be widely applied ; 2) to formulate explicit formulas of spherical functions for concrete examples and analyze the spaces ; 3) to obtain number theoretic interesting quantities for the spaces of symmetric forms and hermitian forms.The most important result is to complete the theory of spherical functions on the space of unramified unitary hermitian matrices.