Locally induced Galois representations with exceptional residual images

Abstract

In this paper, we classify all continuous Galois representations $\rho :\mathrm{Gal}(\bar{Q}/Q)\to {\mathrm{GL}}_2\left({\bar{Q}}_p\right)$ which are unramified outside $\{p,\infty \}$ and locally induced at p, under the assumption that $\bar{\rho }$ 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 ${\rho }_f$ associated to f has exceptional residual image, then ${\rho }_f$ is not locally induced and $a_p\left(f\right)\ne 0$. If ρ is locally induced at p and with exceptional residual image, and furthermore certain subfields of the fixed field of the kernel of $\bar{\rho }$ are assumed to have class numbers prime to p, then ρ has finite image up to a twist.