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 1309.11046Let \(F\) 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.

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.

Reviewer: Davide Lombardo (Orsay)

##### 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 |