Classicality of overconvergent Hilbert eigenforms: case of quadratic residue degrees

Abstract

Let F be a quadratic real field, p be a rational prime inert in F . In this paper, we prove that an overconvergent p-adic Hilbert eigenform for F of small slope is actually a classical Hilbert modular form.Since the left two vertical arrows are isomorphisms, it follows that so is the right one.

Modularity lifting in parallel weight one

“…These results can be stated in terms of the notion of degree of a finite flat group scheme as studied in [14]. In fact, for every finite flat group scheme C corresponding to a point Q ∈ Y rig , this notion can be refined to define partial degrees deg β (C) ∈ [0, 1] ∩ Q, for every β ∈ B (see [37,Def. 3.6]).…”

“…One can then show that ν β (Q) = 1 − deg β (C). Cast in this language, Corollary 3.4 has been proven in both [37] and [28]. [5]).…”

“…To understand the dynamic of the U p Hecke correspondence, we use our results (joint with E. Goren) on the geometry of the Iwahori-level Hilbert modular variety studied in [17], as well as calculations using Breuil-Kisin modules. We should mention that Breuil-Kisin module calculations in this context have also been used by Y. Tian in [37] and first appeared in the work of S. Hattori on canonical subgroups [19].…”

“…This method has been used to prove classicality criteria for overconvergent automorphic forms in several contexts. One can mention [22], [31], [29], [27], and, quite recently, the classicality results in the unramified Hilbert case [28], [37].…”

Classicality for small slope overconvergent automorphic forms on some compact PEL Shimura varieties of type C

“…Recently there has been much progress on also in the geometric theory, using methods that are very different to Coleman's; see [4] for the construction of families and [42], [43] and [47] for classicality results. The method for proving classicality originates from work of Kassaei [23], building on previous work by Buzzard and Taylor [9] on the strong Artin conjecture for two-dimensional representations of Gal(Q/Q), and is in essence a geometric way of analytically continuing the overconvergent form to the whole modular curve (or more generally Shimura variety) of Iwahori level at p. In particular it is entirely different from Coleman's proof, which is cohomological in nature, and instead requires a very explicit understanding of the geometry of the Shimura variety and the geometry of the U p -correspondence.…”