Algebra & Number Theory
- Algebra Number Theory
- Volume 9, Number 4 (2015), 957-980.
Towards local-global compatibility for Hilbert modular forms of low weight
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).
Algebra Number Theory, Volume 9, Number 4 (2015), 957-980.
Received: 24 September 2014
Revised: 6 February 2015
Accepted: 27 March 2015
First available in Project Euclid: 16 November 2017
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 11F41: Automorphic forms on GL(2); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces [See also 14J20]
Secondary: 11F33: Congruences for modular and $p$-adic modular forms [See also 14G20, 22E50] 11F80: Galois representations
Newton, James. Towards local-global compatibility for Hilbert modular forms of low weight. Algebra Number Theory 9 (2015), no. 4, 957--980. doi:10.2140/ant.2015.9.957. https://projecteuclid.org/euclid.ant/1510842342