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

Part of: Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  10 July 2014

PAYMAN L. KASSAEI ,
SHU SASAKI  and
YICHAO TIAN
PAYMAN L. KASSAEI
Affiliation:
Department of Mathematics and Statistics, McGill University, 805 Sherbrooke St. W., Montreal H3A 0B9, QC, Canada; kassaei@math.mcgill.ca
SHU SASAKI
Affiliation:
Fakultat fur Mathematik, Universitat Duisburg-Essen, Thea-Leymann-Strasse 9, 45127 Essen, Germany; s.sasaki.03@cantabgold.net
YICHAO TIAN
Affiliation:
Morningside Center of Mathematics, Chinese Academy of Sciences, 55 Zhong Guan Cun East Road, Beijing, 100190, China; yichaot@math.ac.cn

Abstract

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We 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).

MSC classification

Secondary: 11F33: Congruences for modular and $p$-adic modular forms 11F80: Galois representations
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 2 , 01 July 2014 , e18
Creative Commons
Creative Common License - CCCreative Common License - BY
The online version of this article is published within an Open Access environment subject to the conditions of the Creative Commons Attribution licence .
Copyright
© The Author(s) 2014

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