A constructive approach to p-adic deformations of arithmetic homology
نویسنده
چکیده
Let Γ be an arithmetic group contained in a semigroup S that commensurates Γ and let H be the corresponding Hecke algebra. Given a p-adic analytic family of Banach S-modules varying over an affinoid space Ω and an H-eigenclass ζ in the homology of Γ with coefficients in one member of the family, we give an explicit construction of a p-adic analytic deformation of ζ along some Zariski closed subspace V of Ω. We assume that the S-action is constant modulo p, that some element u ∈ S acts completely continuously on the family, and that ζ is ordinary for the Hecke operator ΓuΓ. Given a p-ordinary Hecke eigenclass in the homology of Γ with coefficients in a finite-dimensional rational representation, our construction yields a family of homology eigenclasses with coefficients in (infinite dimensional) p-adic Banach modules with p-adic weights k. If k is classical, one also obtains homology eigenclasses with coefficients in the finite dimensional highest weight module corresponding to k. A sufficient condition is given for these homology classes to be non-p-torsion. We describe a parallel series of results for cohomology. We give a lower bound for the dimension of V . We describe an example for GL(3)/Q where we can prove that V is at least two dimensional and parametrizes non-p-torsion cohomology classes. 1The author wishes to thank the National Science Foundation for support of this research through NSF grant DMS-0455240. 2000 Mathematics Subject Classification: Primary 11F33, Secondary 11F75.
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