We prove some new cases of local-global compatibility for the Galois
representations associated to Hilbert modular forms of low weight. If
is a totally real
extension of degree
,
we are interested in Hilbert modular forms for
of weight
, with the
and
odd integers and
some but not all
equal to
(the partial weight-one case). Recall that a Hecke eigenform
with such a weight has an associated compatible system
of two-dimensional
-adic
representations of
,
first constructed by Jarvis using congruences to forms of cohomological weight
( for
all
).
One expects that the restriction of the representation
to a decomposition
group
at a
finite place
of
should correspond (under the local Langlands correspondence) to the local factor at
,
, of the automorphic
representation
generated by
.
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
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
has a nonzero monodromy operator. In this paper, we verify local-global
compatibility in many of these ‘missing’ cases, using methods from the
-adic
Langlands programme (including analytic continuation of overconvergent Hilbert
modular forms, maps between eigenvarieties encoding Jacquet–Langlands
functoriality and Emerton’s completed cohomology).
