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 un der 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 lam bda of Galois representations associated to lambda. Finally, under geometri c assumptions (which would be satisfied if one proved that the Galois repre sentations in the family come from Grothmdieck motives), we show that p lam bda is nearly ordinary for the dual parabolic of P. (C) Elsevier, Paris.