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

Abstract

We prove some new cases of local-global compatibility for the Galois representations associated to Hilbert modular forms of low weight. If $F∕\mathbb{Q}$ is a totally real extension of degree $d$, we are interested in Hilbert modular forms for $F$ of weight $\left(k_1,\dots ,k_d,w\right)$, with the $k_i$ and $w$ odd integers and some but not all $k_i$ equal to $1$ (the partial weight-one case). Recall that a Hecke eigenform $f$ with such a weight has an associated compatible system ${\rho }_{f,p}$ of two-dimensional $p$-adic representations of $Gal\left(\overline{F}∕F\right)$, first constructed by Jarvis using congruences to forms of cohomological weight ($k_i\ge 2$ for all $i$).

One expects that the restriction of the representation ${\rho }_{f,p}$ to a decomposition group $D_v$ at a finite place $v@\nmid @p$ of $F$ should correspond (under the local Langlands correspondence) to the local factor at $v$, ${\pi }_{f,v}$, of the automorphic representation ${\pi }_f$ generated by $f$. This expectation is what we refer to as local-global compatibility. For forms of cohomological weight, the compatibility was in most cases verified by Carayol using geometric methods. Combining this result with Jarvis’s construction of Galois representations establishes many cases of local-global compatibility in the partial weight-one situation. However, when ${\pi }_{f,v}$ is a twist of the Steinberg representation, this method establishes a statement weaker that local-global compatibility. The difficulty in this case is to show that the Weil–Deligne representation associated to ${\rho }_{f,p}{|}_{D_v}$ has a nonzero monodromy operator. In this paper, we verify local-global compatibility in many of these ‘missing’ cases, using methods from the $p$-adic Langlands programme (including analytic continuation of overconvergent Hilbert modular forms, maps between eigenvarieties encoding Jacquet–Langlands functoriality and Emerton’s completed cohomology).