论文标题
加洛伊斯的表示形式在一个元素上被淘汰，并且图像适当的图像
Galois representations ramified at one prime and with suitably large image
论文作者
论文摘要
令$ p \ geq 7 $为素数，$ n> 1 $是自然数字。我们表明，存在无限的许多galois表示$ \ varrho：gal（\ bar {\ mathbb {q}}}/\ mathbb {q}）\ rightArrow gl_ {n}（\ mathbb {z} _p）更准确地说，构建的Galois表示的图像包含Mod- $ P^T $还原映射$ sl_n（\ Mathbb {z} _p）\ rightArrow sl_n（\ Mathbb {z}/p^t \ thbb {z}} $ log_p（2^n+1）\ rfloor \ right）+8 $。结果通过纯粹的Galois理论提升结构证明。当$ p \ equiv 1 \ mod {4} $时，我们的结果是有条件的，因为在这种情况下，我们假设Vandiver的猜想非常弱。
Let $p\geq 7$ be a prime and $n>1$ be a natural number. We show that there exist infinitely many Galois representations $\varrho:Gal(\bar{\mathbb{Q}}/\mathbb{Q})\rightarrow GL_{n}(\mathbb{Z}_p)$ which are unramified outside $\{p, \infty\}$ with large image. More precisely, the Galois representations constructed have image containing the kernel of the mod-$p^t$ reduction map $SL_n(\mathbb{Z}_p)\rightarrow SL_n(\mathbb{Z}/p^t\mathbb{Z})$, where $t:=8(n^2-n)\left(3+\lfloor log_p(2^n+1)\rfloor\right)+8$. The results are proven via a purely Galois theoretic lifting construction. When $p\equiv 1\mod{4}$, our results are conditional since in this case, we assume a very weak version of Vandiver's conjecture.
