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Companion forms in parallel weight one
- Part of
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- Journal:
- Compositio Mathematica / Volume 149 / Issue 6 / June 2013
- Published online by Cambridge University Press:
- 10 May 2013, pp. 903-913
- Print publication:
- June 2013
-
- Article
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- You have access
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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 
$F$.
