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Hilbert modular forms modulo $$p$$ and values of extensions between Galois characters. (Formes modulaires de Hilbert modulo $$p$$ et valeurs d’extensions entre caractères galoisiens.) (French. English summary) Zbl 1309.11046
Let $$F$$ be a totally real number field, $$v$$ a place of $$F$$ of characteristic $$p$$, and $$F_v$$ the completion of $$F$$ at $$v$$ (supposed unramified over $$\mathbb{Q}_p$$): the aim of the paper under review is to gain a better understanding of certain smooth representations of $$\mathrm {GL}_2(F_v)$$ with values in $$\mathrm{GL}_2(\overline{\mathbb{F}_p})$$; as the situation is much better understood when $$F_v = \mathbb{Q}_p$$, the authors focus on the opposite case $$F_v \neq \mathbb{Q}_p$$.
In [the first author and V. Paškūnas, Mem. Am. Math. Soc. 1016, 114 p. (2012; Zbl 1245.22010)], families of smooth admissible representations of $$\mathrm {GL}_2(F_v)$$ have been constructed that depend on a very large number of parameters: in general, these parameters have not yet been given a satisfactory interpretation, but the work under consideration sheds some light on their meaning for a specific class of representations (those that are locally reducible and non-split).
More precisely, the representations considered here are of the following form: let $$\overline{\rho}:\mathrm{Gal}(\overline{F}/F) \to \mathrm{GL}_2(\overline{\mathbb{F}_p})$$ be a continuous, irreducible, totally odd representation, and suppose that its restriction to $$\mathrm{Gal}(\overline{F_v}/F_v)$$ is reducible and generic (in the sense of [loc. cit.]); assume furthermore that $$\overline{\rho}$$ is modular, i.e., it arises from the mod-$$p$$ étale cohomology of a tower of Shimura curves. The first main result of the paper associates with $$\overline{\rho}$$ a collection of scalar invariants (denoted $$x(J)$$, where $$J$$ is a family of embeddings of $$k_v$$ in $$\overline{\mathbb{F}_p}$$) with values in $$\overline{\mathbb{F}_p}^\times$$; these $$x(J)$$ are a (usually small) subset of the parameters of [loc. cit.]. The second main theorem of the paper shows that the values of the $$x(J)$$’s can actually be computed explicitly, and only depend on the restriction of $$\overline{\rho}$$ to $$\mathrm{Gal}(\overline{F_v}/F_v)$$ (they are “local”): this is in stark contrast with the situation of [loc. cit.], where smooth admissible representations of $$\mathrm{GL}_2(F_v)$$ are constructed that have fixed $$\mathrm{GL}_2(\mathcal{O}_{F_v})$$-socle, but for which the invariants $$x(J)$$ can take essentially arbitrary values (the crucial difference here being of course the modularity assumption). This second result also implies that the $$x(J)$$’s encode the (non-split) extension between the two characters appearing in $$\mathrm{Gal}\left( \overline{F}_v/F_v\right)$$, thus giving a satisfactory interpretation of these parameters in the case of locally reducible representations.
Finally, an appendix is also included that contains the determination of the mod-$$p$$ semisimplification of the Bushnell-Kutzko types of $$\mathrm{GL}_2$$ and of the units in a quaternion algebra.

##### MSC:
 11F80 Galois representations 11F41 Automorphic forms on $$\mbox{GL}(2)$$; Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces 11S37 Langlands-Weil conjectures, nonabelian class field theory 22E50 Representations of Lie and linear algebraic groups over local fields
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