论文标题
椭圆形曲线的残留galois表示，其图像包含在非切割cartan的正常化中的图像
Residual Galois representations of elliptic curves with image contained in the normaliser of a non-split Cartan
论文作者
论文摘要
众所周知，如果$ p> 37 $是质量数字，而$ e/\ mathbb {q} $是没有复杂乘法的椭圆曲线，那么mod $ p $ galois表示的图像$$ \barρ_{ $ e $的全部是$ \ operatatorName {gl}（e [p]）$的全部，要么是\ emph {contained}的$ \ operatotorname {gl}（gl}（e [p]）$的\ emph {contained}。在本文中，我们表明，当$ p> 1.4 \ times 10^7 $时，$ \barρ_{e，p} $的图像是$ \ operatatorName {gl}（e [p]）$，或\ emph {full}非split cartan cartan子级别的正常人。我们用它来显示以下结果，部分解决了纳吉曼的问题。对于$ d \ geq 1 $，让$ i（d）$表示存在于$ \ m athbb {q} $上定义的椭圆曲线的普莱斯$ p $，并且在不复杂的乘法下，可以承认$ p $等于$ p $ s.egenty定义在一个数量的程度$ \ leq d $的字段上定义。我们表明，对于$ d \ geq 1.4 \ times 10^7 $，我们有 $$ i（d）= \ {p \ text {prime}：p \ leq d-1 \}。 $$
It is known that if $p>37$ is a prime number and $E/\mathbb{Q}$ is an elliptic curve without complex multiplication, then the image of the mod $p$ Galois representation $$ \barρ_{E,p}:\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\rightarrow \operatorname{GL}(E[p]) $$ of $E$ is either the whole of $\operatorname{GL}(E[p])$, or is \emph{contained} in the normaliser of a non-split Cartan subgroup of $\operatorname{GL}(E[p])$. In this paper, we show that when $p>1.4\times 10^7$, the image of $\barρ_{E,p}$ is either $\operatorname{GL}(E[p])$, or the \emph{full} normaliser of a non-split Cartan subgroup. We use this to show the following result, partially settling a question of Najman. For $d\geq 1$, let $I(d)$ denote the set of primes $p$ for which there exists an elliptic curve defined over $\mathbb{Q}$ and without complex multiplication admitting a degree $p$ isogeny defined over a number field of degree $\leq d$. We show that, for $d\geq 1.4\times 10^7$, we have $$ I(d)=\{p\text{ prime}:p\leq d-1\}. $$
