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Companion forms in parallel weight one
Published online by Cambridge University Press: 10 May 2013
Abstract
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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 
$F$.
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