Abstract
Let F be a totally real field and G = GSp(4) / F . In this paper, we show under a weak assumption that, given a Hecke eigensystem λ which is ( p,P)-ordinary for a fixed parabolic P in G, there exists a several-variable p-adic family λ of Hecke eigensystems (all of them ( p,P)-nearly ordinary) which contains λ. The assumption is that λ is cohomological for a regular coefficient system. If F = Q, the number of variables is three. Moreover, in this case, we construct the three-variable p-adic family ρ λ of Galois representations associated to λ . Finally, under geometric assumptions (which would be satisfied if one proved that the Galois representations in the family come from Grothendieck motives), we show that ρ λ is nearly ordinary for the dual parabolic of P.
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