### Several variable p-adic families of Siegel-Hilbert cusp eigensystems and their Galois representations, by Jacques Tilouine and Eric Urban

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 underline{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. If F=Q, the number of
variables is three. Moreover, in this case, we construct the three variable
p-adic family rho_{underline{lambda}} of Galois representations associated to
underline{lambda}. 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 rho_{underline{lambda}} is nearly
ordinary for the dual parabolic of P. This text is an updated version of our
first preprint (issued in the "Prepublication de l'universite Paris-Nord")
and will appear in the "Annales Scientifiques de l' E N S".

Jacques Tilouine <tilouine@math.univ-paris13.fr>
Eric Urban <urban@math.ucla.edu>