论文标题
具有特殊残留图像的本地引起的Galois表示
Locally induced Galois representations with exceptional residual images
论文作者
论文摘要
在本文中，我们将所有连续的Galois表示$ρ：\ MATHRM {gal}（\ edimline {\ MathBf {Q}}}/\ Mathbf {q}）\ to \ m atrm {gl} _2 $ \ {p，\ iffty \} $，并以$ p $为$ p $的本地引起的，假设$ \overlineρ$是例外的，即订单prime to prime to $ p $。我们证明了两个结果。如果$ f $是一级cuspidal特征形式，而与$ f $相关的$ p $ - adic galois表示$ρ_f$具有出色的残留图像，则$ρ_f$不是本地引起的，而$ a_p（f）\ neq 0 $。如果$ρ$以$ P $为本地引起，并且具有出色的剩余图像，此外，假定$ \overlineρ$的固定字段的某些子字段被认为具有班级数字为$ p $，则$ρ$具有有限的图像。
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.
