Vol. 9, No. 4, 2015

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

James Newton

Vol. 9 (2015), No. 4, 957–980

We prove some new cases of local-global compatibility for the Galois representations associated to Hilbert modular forms of low weight. If F is a totally real extension of degree d, we are interested in Hilbert modular forms for F of weight (k1,,kd,w), with the ki and w odd integers and some but not all ki equal to 1 (the partial weight-one case). Recall that a Hecke eigenform f with such a weight has an associated compatible system ρf,p of two-dimensional p-adic representations of Gal( F¯F), first constructed by Jarvis using congruences to forms of cohomological weight (ki 2 for all i).

One expects that the restriction of the representation ρf,p to a decomposition group Dv at a finite place v@@p of F should correspond (under the local Langlands correspondence) to the local factor at v, πf,v, of the automorphic representation πf 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 πf,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 ρf,p|Dv 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).

Hilbert modular forms, Galois representations, local-global compatibility
Mathematical Subject Classification 2010
Primary: 11F41
Secondary: 11F33, 11F80
Received: 24 September 2014
Revised: 6 February 2015
Accepted: 27 March 2015
Published: 30 May 2015
James Newton
Department of Mathematics
Imperial College London
South Kensington Campus
London SW7 2AZ
United Kingdom