arXiv 1809.025842018-09math.NT

椭圆曲线上的复乘伽罗瓦表示

Galois representations attached to elliptic curves with complex multiplication

Álvaro Lozano-Robledo

备注：: 55 pages. Updated after referee reports. To appear in "Algebra and Number Theory"
发表信息：: Alg. Number Th. 16 (2022) 777-837
DOI：: 10.2140/ant.2022.16.777

摘要：

本文的目标是给出一个明确的分类可能的$p$ -adic伽罗瓦表示，附加到椭圆曲线$E$与CM定义在$\mathbb{Q}(j(E))$。更准确地说，令$K$是虚二次域，并且令$\mathcal{O}_{K,f}$是导体$f\geq 1$在$K$中的阶。令$E$是CM乘$\mathcal{O}_{K,f}$的椭圆曲线，使得$E$由$\mathbb{Q}(j(E))$上的模型定义。令$p\geq 2$是素数，令$G_{\mathbb{Q}(j(E))}$是$\mathbb{Q}(j(E))$的绝对伽罗瓦群，令$ρ_{E,p^\infty}\colon G_{\mathbb{Q}(j(E))}\to \operatorname{GL}(2,\mathbb{Z}_p)$是与Tate模块$T_p(E)$上的伽罗瓦作用相关联的伽罗瓦表示。然后，我们的目标是明确地描述可以作为$ρ_{E,p^\infty}$的图像出现的$\operatorname{GL}(2,\mathbb{Z}_p)$的组，直到共轭，对于任意阶$\mathcal{O}_{K,f}$。

英文摘要：

The goal of this article is to give an explicit classification of the possible $p$-adic Galois representations that are attached to elliptic curves $E$ with CM defined over $\mathbb{Q}(j(E))$. More precisely, let $K$ be an imaginary quadratic field, and let $\mathcal{O}_{K,f}$ be an order in $K$ of conductor $f\geq 1$. Let $E$ be an elliptic curve with CM by $\mathcal{O}_{K,f}$, such that $E$ is defined by a model over $\mathbb{Q}(j(E))$. Let $p\geq 2$ be a prime, let $G_{\mathbb{Q}(j(E))}$ be the absolute Galois group of $\mathbb{Q}(j(E))$, and let $ρ_{E,p^\infty}\colon G_{\mathbb{Q}(j(E))}\to \operatorname{GL}(2,\mathbb{Z}_p)$ be the Galois representation associated to the Galois action on the Tate module $T_p(E)$. The goal is then to describe, explicitly, the groups of $\operatorname{GL}(2,\mathbb{Z}_p)$ that can occur as images of $ρ_{E,p^\infty}$, up to conjugation, for an arbitrary order $\mathcal{O}_{K,f}$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

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