Gespeichert in:
| 1. Verfasser: | |
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| Format: | Preprint |
| Veröffentlicht: |
2022
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| Schlagworte: | |
| Online-Zugang: | https://arxiv.org/abs/2205.11562 |
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Inhaltsangabe:
- In this paper, we classify all continuous Galois representations $ρ:\mathrm{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})\to \mathrm{GL}_2(\overline{\mathbf{Q}}_p)$ which are unramified outside $\{p,\infty\}$ and locally induced at $p$, under the assumption that $\overlineρ$ is exceptional, that is, has image of order prime to $p$. We prove two results. If $f$ is a level one cuspidal eigenform and one of the $p$-adic Galois representations $ρ_f$ associated to $f$ has exceptional residual image, then $ρ_f$ is not locally induced and $a_p(f)\neq 0$. If $ρ$ is locally induced at $p$ and with exceptional residual image, and furthermore certain subfields of the fixed field of the kernel of $\overlineρ$ are assumed to have class numbers prime to $p$, then $ρ$ has finite image up to a twist.