Algebra & Number Theory

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

James Newton

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at msp.org/ant.

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

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).

Article information

Source
Algebra Number Theory, Volume 9, Number 4 (2015), 957-980.

Dates
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
https://projecteuclid.org/euclid.ant/1510842342

Digital Object Identifier
doi:10.2140/ant.2015.9.957

Mathematical Reviews number (MathSciNet)
MR3352826

Zentralblatt MATH identifier
1369.11035

Subjects
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

Keywords
Hilbert modular forms Galois representations local-global compatibility

Citation

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


Export citation

References

  • A. Ash and G. Stevens, “$p$-adic deformations of arithmetic cohomology”, preprint, 2008, hook http://www2.bc.edu/~ashav/Papers/Ash-Stevens-Oct-07-DRAFT-copy.pdf \posturlhook.
  • J. Bellaïche and G. Chenevier, Families of Galois representations and Selmer groups, Astérisque 324, Société Mathématique de France, Paris, 2009.
  • D. Blasius and J. D. Rogawski, “Motives for Hilbert modular forms”, Invent. Math. 114:1 (1993), 55–87.
  • K. Buzzard, “On $p$-adic families of automorphic forms”, pp. 23–44 in Modular curves and abelian varieties, edited by J. Cremona et al., Progr. Math. 224, Birkhäuser, Basel, 2004.
  • K. Buzzard, “Eigenvarieties”, pp. 59–120 in $L$-functions and Galois representations, edited by D. Burns et al., London Math. Soc. Lecture Note Ser. 320, Cambridge University Press, 2007.
  • H. Carayol, “Sur la mauvaise réduction des courbes de Shimura”, Compositio Math. 59:2 (1986), 151–230.
  • H. Carayol, “Sur les représentations $l$-adiques associées aux formes modulaires de Hilbert”, Ann. Sci. École Norm. Sup. $(4)$ 19:3 (1986), 409–468.
  • G. Chenevier, “Familles $p$-adiques de formes automorphes pour ${\rm GL}\sb n$”, J. Reine Angew. Math. 570 (2004), 143–217.
  • G. Chenevier, “Une application des variétés de Hecke des groups unitaires”, preprint, 2009, hook http://gaetan.chenevier.perso.math.cnrs.fr/articles/famgal.pdf \posturlhook.
  • B. Conrad, “Irreducible components of rigid spaces”, Ann. Inst. Fourier $($Grenoble$)$ 49:2 (1999), 473–541.
  • M. Emerton, “Jacquet modules of locally analytic representations of $p$-adic reductive groups, I: Construction and first properties”, Ann. Sci. École Norm. Sup. $(4)$ 39:5 (2006), 775–839.
  • M. Emerton, “On the interpolation of systems of eigenvalues attached to automorphic Hecke eigenforms”, Invent. Math. 164:1 (2006), 1–84.
  • M. Emerton, “Locally analytic vectors in representations of locally $p$-adic analytic groups”, preprint, 2011, hook http://www.math.uchicago.edu/~emerton/pdffiles/analytic.pdf \posturlhook. To appear in Memoirs of the AMS.
  • D. Goldfeld and J. Hundley, Automorphic representations and $L$-functions for the general linear group, I, Cambridge Studies in Advanced Mathematics 129, Cambridge University Press, 2011.
  • H. Jacquet and R. P. Langlands, Automorphic forms on ${\rm GL}(2)$, Lecture Notes in Mathematics 114, Springer, Berlin-New York, 1970.
  • F. Jarvis, “On Galois representations associated to Hilbert modular forms”, J. Reine Angew. Math. 491 (1997), 199–216.
  • A. Jorza, Crystalline representations for GL(2) over quadratic imaginary fields, Ph.D. thesis, Princeton University, 2010, hook http://search.proquest.com/docview/522118786 \posturlhook.
  • P. L. Kassaei, “Modularity lifting in parallel weight one”, J. Amer. Math. Soc. 26:1 (2013), 199–225.
  • P. L. Kassaei, S. Sasaki, and Y. Tian, “Modularity lifting results in parallel weight one and applications to the Artin conjecture: the tamely ramified case”, Forum Math. Sigma 2 (2014), 58 (electronic).
  • M. Kisin and K. F. Lai, “Overconvergent Hilbert modular forms”, Amer. J. Math. 127:4 (2005), 735–783.
  • D. Loeffler, “Overconvergent algebraic automorphic forms”, Proc. Lond. Math. Soc. $(3)$ 102:2 (2011), 193–228.
  • M. Luu, Deformation theory and local-global compatibility of Langlands correspondences, Memoirs of the AMS 1123, Amer. Math. Soc., Providence, RI, online publication February 2015.
  • R. Moy and J. Specter, “There exist non-CM Hilbert modular forms of partial weight 1”, preprint, 2015.
  • J. Newton, “Completed cohomology of Shimura curves and a $p$-adic Jacquet–Langlands correspondence”, Math. Ann. 355:2 (2013), 729–763.
  • J. Newton, “Level raising for $p$-adic Hilbert modular forms”, preprint, 2014. http://msp.org/idx/arx/1409.6533arXiv 1409.6533
  • A. G. M. Paulin, “Local to global compatibility on the eigencurve”, Proc. Lond. Math. Soc. $(3)$ 103:3 (2011), 405–440.
  • V. Pilloni and B. Stroh, “Surconvergence et classicité: le cas Hilbert”, preprint, 2011, hook http://perso.ens-lyon.fr/vincent.pilloni/surconv_hilbert.pdf \posturlhook.
  • V. Pilloni and B. Stroh, “Surconvergence, ramification et modularité”, preprint, 2013, hook http://perso.ens-lyon.fr/vincent.pilloni/Artinfinal.pdf \posturlhook.
  • K. A. Ribet, “Galois representations attached to eigenforms with Nebentypus”, pp. 17–51 in Modular functions of one variable, V (Univ. Bonn, 1976), edited by J.-P. Serre and D. Zagier, Lecture Notes in Mathematics 601, Springer, Berlin, 1977.
  • J. D. Rogawski and J. B. Tunnell, “On Artin $L$-functions associated to Hilbert modular forms of weight one”, Invent. Math. 74:1 (1983), 1–42.
  • S. Sasaki, “Integral models of Hilbert modular varieties in the ramified case, deformations of modular Galois representations, and weight one forms”, preprint, 2014, hook http://www.cantabgold.net/users/s.sasaki.03/hmv1-3-14.pdf \posturlhook.
  • R. Taylor, “On Galois representations associated to Hilbert modular forms”, Invent. Math. 98:2 (1989), 265–280.
  • R. Taylor, “On Galois representations associated to Hilbert modular forms, II”, pp. 185–191 in Elliptic curves, modular forms & Fermat's last theorem (Hong Kong, 1993), edited by J. Coates, Ser. Number Theory I, Int. Press, Cambridge, MA, 1995.
  • Y. Tian and L. Xiao, “$p$-adic cohomology and classicality of overconvergent Hilbert modular forms”, preprint, 2013.