Abstract. Soit L une extension finie non ramifiée de ℚ p . On s'intéresse au problème de savoir si certaines des représentations de GL 2 ( L ) $\operatorname{GL}_2(L)$ sur 𝔽 p ¯ $\overline{\mathbb {F}_{p}}$ associées par Breuil et Paškūnas à une représentation de Gal ( ℚ p ¯ / L ) ${\operatorname{Gal}}(\overline{\mathbb {Q}_{p}}/L)$ de dimension 2 sur 𝔽 p ¯ $\overline{\mathbb {F}_{p}}$ générique peuvent apparaître dans des espaces de formes automorphes.

We exhibit algorithms to compute systems of Hecke eigenvalues for spaces of Hilbert modular forms over a totally real field. We provide many explicit examples as well as applications to modularity and Galois representations.

A generalization of Serre's Conjecture asserts that if F is a totally real field, then certain characteristic p representations of Galois groups over F arise from Hilbert modular forms. Moreover, it predicts the set of weights of such forms in terms of the local behaviour of the Galois representation at primes over p. This characterization of the weights, which is formulated using p-adic Hodge theory, is known under mild technical hypotheses if p > 2. In this paper we give, under the assumption that p is unramified in F, a conjectural alternative description for the set of weights. Our approach is to use the Artin-Hasse exponential and local class field theory to construct bases for local Galois cohomology spaces in terms of which we identify subspaces that should correspond to ones defined using p-adic Hodge theory. The resulting conjecture amounts to an explicit description of wild ramification in reductions of certain crystalline Galois representations. It enables the direct computation of the set of Serre weights of a Galois representation, which we illustrate with numerical examples. A proof of this conjecture has been announced by Calegari, Emerton, Gee and Mavrides.

We explain how the work of Johnson-Leung and Roberts on lifting Hilbert modular forms for real quadratic fields to Siegel modular forms can be adapted to imaginary quadratic fields. For this, we use archimedean results from Harris, Soudry and Taylor and replace the global arguments of Roberts by the non-vanishing result of Takeda. As an application of our lifting result, we exhibit an abelian surface B defined over Q, which is not a restriction of scalars of an elliptic curve and satisfies the paramodularity Conjecture of Brumer and Kramer.