On local Galois representations associated to ordinary Hilbert modular forms

2013 ◽  
Vol 142 (3-4) ◽  
pp. 513-524 ◽  
Author(s):  
Baskar Balasubramanyam ◽  
Eknath Ghate ◽  
Vinayak Vatsal
Keyword(s):  
Modular Forms ◽  
Galois Representations ◽  
Hilbert Modular Forms ◽  
Hilbert Modular

scholarly journals UNRAMIFIEDNESS OF GALOIS REPRESENTATIONS ATTACHED TO HILBERT MODULAR FORMS MOD  OF WEIGHT 1

2018 ◽  
Vol 19 (2) ◽  
pp. 281-306 ◽  
Author(s):  
Mladen Dimitrov ◽  
Gabor Wiese
Keyword(s):  
Modular Forms ◽  
Galois Representations ◽  
Galois Representation ◽  
Important Case ◽  
Hilbert Modular Forms ◽  
Hilbert Modular

The main result of this article states that the Galois representation attached to a Hilbert modular eigenform defined over $\overline{\mathbb{F}}_{p}$ of parallel weight 1 and level prime to $p$ is unramified above $p$. This includes the important case of eigenforms that do not lift to Hilbert modular forms in characteristic 0 of parallel weight 1. The proof is based on the observation that parallel weight 1 forms in characteristic $p$ embed into the ordinary part of parallel weight $p$ forms in two different ways per prime dividing $p$, namely via ‘partial’ Frobenius operators.

Lattices in Filtered (φ, N)-modules

2012 ◽  
Vol 11 (3) ◽  
pp. 659-693 ◽  
Author(s):  
Tong Liu
Keyword(s):  
Modular Forms ◽  
Galois Representations ◽  
Hilbert Modular Forms ◽  
Langlands Correspondence ◽  
Hilbert Modular ◽  
Local Langlands Correspondence

AbstractLet p be a prime. We construct and study integral and torsion invariants, such as integral and torsion Weil–Deligne representations, associated to potentially semi-stable representations and torsion potentially semi-stable representations respectively. As applications, we prove the compatibility between local Langlands correspondence and Fontaine's construction for Galois representations attached to Hilbert modular forms, and Néron–Ogg–Shafarevich criterion of finite level for potentially semi-stable representations.

scholarly journals Ramification of the Eigencurve at Classical RM Points

2019 ◽  
Vol 72 (1) ◽  
pp. 57-88
Author(s):  
Adel Betina
Keyword(s):  
Modular Forms ◽  
Quadratic Field ◽  
Galois Representations ◽  
Theta Series ◽  
Base Change ◽  
Hilbert Modular Forms ◽  
Deformation Problem ◽  
Ramification Index ◽  
Hilbert Modular ◽  
Real Quadratic

AbstractJ. Bellaïche and M. Dimitrov showed that the $p$-adic eigencurve is smooth but not étale over the weight space at $p$-regular theta series attached to a character of a real quadratic field $F$ in which $p$ splits. In this paper we prove the existence of an isomorphism between the subring fixed by the Atkin–Lehner involution of the completed local ring of the eigencurve at these points and a universal ring representing a pseudo-deformation problem. Additionally, we give a precise criterion for which the ramification index is exactly 2. We finish this paper by proving the smoothness of the nearly ordinary and ordinary Hecke algebras for Hilbert modular forms over $F$ at the overconvergent cuspidal Eisenstein points, being the base change lift for $\text{GL}(2)_{/F}$ of these theta series. Our approach uses deformations and pseudo-deformations of reducible Galois representations.

Overconvergent Families of Siegel–Hilbert Modular Forms

2015 ◽  
Vol 67 (4) ◽  
pp. 893-922 ◽  
Author(s):  
Chung Pang Mok ◽  
Fucheng Tan
Keyword(s):  
Modular Forms ◽  
Automorphic Forms ◽  
Galois Representations ◽  
Hilbert Modular Forms ◽  
Hilbert Modular

AbstractWe construct one-parameter families of overconvergent Siegel-Hilbert modular forms. This result has applications to the construction of Galois representations for automorphic forms of noncohomological weights.

scholarly journals Modularity lifting results in parallel weight one and applications to the Artin conjecture: the tamely ramified case

2014 ◽  
Vol 2 ◽  
Author(s):  
PAYMAN L. KASSAEI ◽  
SHU SASAKI ◽  
YICHAO TIAN
Keyword(s):  
Analytic Continuation ◽  
General Result ◽  
Modular Forms ◽  
Galois Representations ◽  
Hilbert Modular Forms ◽  
Hilbert Modular ◽  
Weight One ◽  
Totally Real ◽  
Totally Real Fields ◽  
Finite Slope

AbstractWe extend the modularity lifting result of P. Kassaei (‘Modularity lifting in parallel weight one’,J. Amer. Math. Soc.26 (1) (2013), 199–225) to allow Galois representations with some ramification at $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}p$. We also prove modularity mod 5 of certain Galois representations. We use these results to prove new cases of the strong Artin conjecture over totally real fields in which 5 is unramified. As an ingredient of the proof, we provide a general result on the automatic analytic continuation of overconvergent $p$-adic Hilbert modular forms of finite slope which substantially generalizes a similar result in P. Kassaei (‘Modularity lifting in parallel weight one’, J. Amer. Math. Soc.26 (1) (2013), 199–225).

Sign in / Sign up

Export Citation Format

Share Document