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 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"

Big image of Galois representations associated with finite slope pp-adic families of modular forms

We consider the Galois representation associated with a finite slope $p$-adic family of modular forms. We prove that the Lie algebra of its image contains a congruence Lie subalgebra of a non-trivial level. We describe the largest such level in terms of the congruences of the family with $p$-adic CM forms.Comment: 23 pages; revision of Section 2 (see Remark 2.4) and improvement of Proposition 4.14, plus minor changes. Published in "Elliptic Curves, Modular Forms and Iwasawa Theory. In Honour of John H. Coates' 70th Birthday, Cambridge, UK, March 2015", Springer Proceedings in Mathematics & Statistics, Vol. 188, 201

Conjecture de type de Serre et formes compagnons pour GSp_4

We present a Serre-type conjecture on the modularity of four-dimensional symplectic mod p Galois representations. We assume that the Galois representation is irreducible and odd (in the symplectic sense). The modularity condition is formulated using the etale and the algebraic de Rham cohomology of Siegel modular varieties of level prime to p. We concentrate on the case when the Galois representation is ordinary at p and we give a corresponding list of Serre weights. When the representation is moreover tamely ramified at p, we conjecture that all weights of this list are modular, otherwise we describe a subset of weights on the list that should be modular. We propose a construction of de Rham cohomology classes using the dual BGG complex, which should realise some of these weights.Comment: 36 page

BIG IMAGE OF GALOIS REPRESENTATIONS AND CONGRUENCE IDEALS

2. Galois representations associated to Siegel modular forms 2 3. Fullness of the image for Galois representations in GSp(4) 3 3.1. Irreducibility and open image