论文标题
在一些$ p $ - adid galois表示和表格班级组中
On some $p$-adic Galois representations and form class groups
论文作者
论文摘要
令$ k $是带有整数$ \ MATHCAL {O} _K $的判别$ d_k $的假想二次字段。当$ k $不同于$ \ mathbb {q}（\ sqrt {-1}）$和$ \ mathbb {q}（\ sqrt {-3}）$时，我们考虑了椭圆形曲线$ e_k $ at $ j（e_k k）= j（e_k）= j（e_k）= j（\ m mathcal paind）的某些特定模型， $ \ mathbb {q}（j（e_k））$。在本文中，对于每个正整数$ n $，我们将$ n $ torsion in $ e_k $ $ n $ torsion积分生成的$ \ mathbb {q} $的扩展字段与$ e class Field $ k _ {（n）} $ k $ k $ modulo $ n \ n \ nathcal {o n \ mathcal {o。通过使用此结果，我们调查了$ e_k $ $ e_k $的Prime $ P $的图像，就班级字段理论而言。其次，我们构建了判别$ d_k $和级别$ n $的确定表单类别组，该组是同构至$ \ mathrm {gal}（k _ {（n）}/\ mathbb {q}）$。
Let $K$ be an imaginary quadratic field of discriminant $d_K$ with ring of integers $\mathcal{O}_K$. When $K$ is different from $\mathbb{Q}(\sqrt{-1})$ and $\mathbb{Q}(\sqrt{-3})$, we consider a certain specific model for the elliptic curve $E_K$ with $j(E_K)=j(\mathcal{O}_K)$ which is defined over $\mathbb{Q}(j(E_K))$. In this paper, for each positive integer $N$ we compare the extension field of $\mathbb{Q}$ generated by the coordinates of $N$-torsion points on $E_K$ with the ray class field $K_{(N)}$ of $K$ modulo $N\mathcal{O}_K$. By using this result we investigate the image of a $p$-adic Galois representation attached to $E_K$ for a prime $p$, in terms of class field theory. Second, we construct the definite form class group of discriminant $d_K$ and level $N$ which is isomorphic to $\mathrm{Gal}(K_{(N)}/\mathbb{Q})$.
