Several-variable p-adic families of Siegel-Hilbert cusp eigensystems and their Galois representations

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 lambda which is (p, P)-ordinary for a fixed parabolic P in G, there exists a several-variable p-adic family lambda of Hecke eigensystems (all of them (p,P)-nearly ordinary) which contains lambda. The assumption is that lambda is cohomological for a regular coefficient system. Lf F = Q, the number of variables is three. Moreover, in this case, we construct the three-variable p-adic family p lambda of Galois representations associated to lambda. Finally, under geometric assumptions (which would be satisfied if one proved that the Galois representations in the family come from Grothmdieck motives), we show that p lambda is nearly ordinary for the dual parabolic of P. (C) Elsevier, Paris.