Jacques Tilouine

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 λ which is (p,P)-ordinary for a fixed parabolic P in G, there exists a several-variable p-adic family λ of Hecke eigensystems (all of them (p,P)-nearly ordinary) which contains λ. The assumption is that λ 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 ρλ of Galois representations associated to λ. 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 ρλ is nearly ordinary for the dual parabolic of P.

Hecke Algebras and the Gorenstein Property

The goal of this paper is to show the importance of the Gorenstein property for the Hecke algebra and its relation with the local freeness of the cohomology of modular curves as a module over the Hecke algebra.

Nearly ordinary rank four Galois representations and

This paper is devoted to the proof of two results. The first was conjectured in 1994 by the author. It concerns the identity, under certain assumptions, of the universal deformation ring of p-nearly ordinary Galois representations and a local component of the universal nearly ordinary Hecke algebra in the sense of Hida. The other, which relies on the first, concerns the modularity of certain abelian surfaces. More precisely, one can associate to certain irreducible abelian surfaces defined over the rationals overconvergent p-adic cusp eigenforms. The question of whether these forms are classical is not studied in this paper.

p

We consider the Galois representation associated with a finite slope

-adic Siegel modular forms

p

Big Image of Galois Representations Associated with Finite Slope p -adic Families of Modular Forms

-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 Aspects of Modular Forms

p

Anti-cyclotomic Katz p-adic L-functions and congruence modules

-adic CM forms.We prove that the Lie algebra of the image of the Galois representation associated with a finite slope family of modular forms contains a congruence subalgebra of a certain level. We interpret this level in terms of congruences with CM forms.

Deformations of Galois representations and Hecke algebras

In this short note we partially answer a question of Fukaya and Kato by constructing a

On the anticyclotomic main conjecture for CM fields.

q