Modularity lifting in parallel weight one

Abstract

We prove an analogue of a modularity lifting result of Buzzard and Taylor for totally real fields in which p p is unramified. This can be used to prove certain cases of the strong Artin conjecture over totally real fields.

Towards local-global compatibility for Hilbert modular forms of low weight

“…Applying [18,Theorem 7.10], we glue the g S into a classical Hilbert modular eigenform g ∈ H 0 (M (n, q) L , ω κ )…”

“…We will say a few words about the hypotheses in Theorem 1.4. Hypotheses (1) and (2) are satisfied for all but finitely many p. Hypothesis (1) appears because we use results of Kassaei [18] on analytic continuation and gluing of overconvergent Hilber modular forms. At least for forms of parallel weight, results have been announced by Pilloni, Stroh [27] and Sasaki [30] which apply without restriction on the ramification of p, so it seems reasonable to hope that this hypothesis could be removed in future.…”

“…We note that Luu [21, §3.2] has also recently described an approach to proving some cases of local-global compatibility for Hilbert modular forms of low weight, when π satisfies an ordinarity hypothesis at places dividing p. We will use the same notation as in the previous subsection. Using the assumption that ρ v is unramified, Luu applies a modularity lifting theorem to produce an ordinary p-adic Hilbert modular form g with level prime to v and the same system of Hecke eigenvalues (outside v) as π, and imposes a hypothesis that amounts to ruling out the existence of g. It may also be possible to show that g is classical (hence obtaining a contradiction) using a variant of the methods applied in the parallel weight 1 case (as in, for example, [18]), but it is not obvious to the author that these methods can be easily applied. One obstacle is that, unlike the parallel weight 1 case, not all the p-stabilisations of the newform generating π are ordinary.…”

A conditional construction of Artin representations for real analytic Siegel cusp forms of weight (2,1)

“…(See [41], [37] for any Hilbert cusp form with parallel weight one. For partial results on the automorphy of Galois representations, see [44], [19], [11]. )…”

Modularity lifting results in parallel weight one and applications to the Artin conjecture: the tamely ramified case

“…In [19], the first-named author proved a generalization of the main result of Buzzard-Taylor [6] over a totally real field in which p is unramified. In [27], Pilloni proved a generalization of this result in the case where p is slightly ramified in F .…”