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

TL;DR

This paper constructs multi-variable p-adic families of Siegel-Hilbert cusp eigensystems and their associated Galois representations over totally real fields, under certain cohomological and geometric assumptions.

Contribution

It establishes the existence of several variable p-adic families of Hecke eigensystems containing a given eigensystem, and constructs associated Galois representations in the case of F=Q.

Findings

Existence of multi-variable p-adic families of eigensystems.

Construction of Galois representations for these families.

Nearly ordinary property of Galois representations under geometric assumptions.

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 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".