Product Of P-adic Symmetric Spaces

p-adic symmetric spaces play the role of complex upper half-plane in the p-adic analytic world and they are rigid spaces.In this paper we generalize a series of known results for p-adic symmetry space to the product of p-adic symmetric spaces Π_i=1^r（?）^di.Let Γ be a torsion free and discrete subgroup of Π_i=1^r（PGL）_di+1（K）,we prove the exis-tence of covering spectral sequence for de Rham cohomology group of quotient spaceΓ\Π_i=1^r（?）^di.Moreover if Γ is cocompact,we also prove the existence of Hodge-to-de Rham spectral sequence.