The Jacquet–Langlands correspondence for overconvergent Hilbert modular forms

Abstract

We use results by Chenevier to interpolate the classical Jacquet-Langlands correspondence for Hilbert modular forms, which gives us an extension of Chenevier's results to totally real fields. From this we obtain an isomorphisms between eigenvarieties attached Hilbert modular forms and those attached to modular forms on a totally definite quaternion algebra over a totally real field of even degree. a In general Chenevier proves that one gets a isomorphism onto the d-new 'part' of the eigenvariety.

Modularity of trianguline Galois representations

“…We then turn to the automorphic theory we will need. We prove that so-called twist classical points are very Zariski dense in the eigenvariety X D × , which permits us to interpolate the Jacquet-Langlands correspondence to extended eigenvarieties and permits us to conclude that X D × is reduced (extending the results of [Bir19] and [Che05]). We also study the cyclic base change morphism X GL2 /Q → X GL2 /F of [JN19a]; when F is totally real and [F : Q] is prime to p, we show that x ∈ X GL2 /F is in the image if and only if it is fixed by Gal(F/Q).…”

“…The classical Jacquet-Langlands correspondence lets us transfer automorphic forms between GL 2 and quaternionic algebraic groups. Over Q, this correspondence was interpolated in [Che05] to give a closed immersion of eigencurves X rig D × /Q → X rig GL2 /Q ; this interpolation was given for general totally real fields in [Bir19]. We give the corresponding result for extended eigenvarieties.…”

“…However, there are a number of technical complications. For example, to carry out some preliminary reductions, we first prove a version of the Jacquet-Langlands correspondence on eigenvarieties extending the construction of [Bir19], and we characterize the image of the cyclic base change morphism X GL2 /Q → X GL2 /F of [JN19a]. The latter uses the construction of an auxiliary "Gal(F/Q)-fixed" eigenvariety, which may be of independent interest.…”

Cohomology of $(φ,Γ)$-modules over pseudorigid spaces

“…When H is a totally definite quaternion algebra over a totally real field, split at p, a similar argument shows that X rig G contains a Zariski very dense set of classical points. Moreover, the p-adic Jacquet-Langlands correspondance of [Che05], [Bir19] can be extended to the pseudorigid setting. This identifies each irreducible component of a quaternionic eigenvariety with an irreducible component of an eigenvariety for Hilbert modular forms; it follows that a Galois determinant can be pulled back to X G , and it corresponds to a trianguline representation at a dense set of points of X rig G .…”

2‐adic slopes of Hilbert modular forms over Q(5)

“…As is usual, instead of working directly with the space of overconvergent Hilbert modular forms, we will instead work with spaces of overconvergent quaternionic modular forms as the geometry is simpler. Specifically, by [2, Theorem 1], if we let be the unique quaternion algebra over ramifying only at the two infinite places of , then the eigenvariety associated to Hilbert modular forms (as defined by [1]) is isomorphic to the eigenvariety associated to these quaternionic forms. Therefore, since we are only interested in slopes, there is no loss in working with overconvergent quaternionic forms.…”