Let $n>1$, $e\ge 0$ and a prime number $p\ge 2^{n+2+2e}+3$, such that the index of regularity of p is ≤e. We show that there are infinitely many irreducible Galois representations $\rho :\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\to {\mathrm{GL}}_n({\mathbb{Q}}_p)$ unramified at all primes $l\ne p$. Furthermore, these representations are shown to have image containing a fixed finite index subgroup of ${\mathrm{SL}}_n({\mathbb{Z}}_p)$. Such representations are constructed by lifting suitable residual representations $\overline{\rho }$ with image in the diagonal torus in ${\mathrm{GL}}_n({\mathbb{F}}_p)$, for which the global deformation problem is unobstructed.

中文翻译：

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构造伽罗瓦表示法一枝独秀

让 $ñ>\mathrm{1个}$， $Ë\ge 0$ 和素数 $p\ge 2^{ñ+2+2Ë}+3$，使得规律性的指数p是≤ Ë。我们证明了有无限多个不可约的伽罗瓦表示$\rho ：\mathrm{加尔}（\overline{\mathbb{问}}/\mathbb{问}）\to {\mathrm{GL}}_{ñ}（{\mathbb{问}}_p）$ 在所有素数均未分叉 $升\ne p$。此外，这些表示形式显示为包含固定的有限索引子组的图像。${\mathrm{SL}}_{ñ}（{\mathbb{ž}}_p）$。通过提升适当的残差表示来构造此类表示$\overline{\rho }$ 与图像在对角线环面中 ${\mathrm{GL}}_{ñ}（{\mathbb{F}}_p）$，其全局变形问题不受阻碍。