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Companion forms in parallel weight one

Part of: Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  10 May 2013

Toby Gee  and
Payman Kassaei
Toby Gee
Affiliation:
Department of Mathematics, Imperial College London, London SW7 2AZ, UK email toby.gee@imperial.ac.uk
Payman Kassaei
Affiliation:
Department of Mathematics, King’s College London, London WC2R 2LS, UK email payman.kassaei@kcl.ac.uk
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Abstract

Let $p\gt 2$ be prime, and let $F$ be a totally real field in which $p$ is unramified. We give a sufficient criterion for a $\mathrm{mod} \hspace{0.167em} p$ Galois representation to arise from a $\mathrm{mod} \hspace{0.167em} p$ Hilbert modular form of parallel weight one, by proving a ‘companion forms’ theorem in this case. The techniques used are a mixture of modularity lifting theorems and geometric methods. As an application, we show that Serre’s conjecture for $F$ implies Artin’s conjecture for totally odd two-dimensional representations over $F$.

Keywords

companion formsparallel weight oneSerre’s conjecturemod p Hilbert modular forms

MSC classification

Secondary: 11F33: Congruences for modular and $p$-adic modular forms 11F41: Automorphic forms on $(2)$; Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces 11F80: Galois representations
Type
Research Article
Information
Compositio Mathematica , Volume 149 , Issue 6 , June 2013 , pp. 903 - 913
Copyright
© The Author(s) 2013 

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