论文标题
复杂乘法
Entanglement in the family of division fields of elliptic curves with complex multiplication
论文作者
论文摘要
For every elliptic curve $E$ which has complex multiplication (CM) and is defined over a number field $F$ containing the CM field $K$, we prove that the family of $p^{\infty}$-division fields of $E$, with $p \in \mathbb{N}$ prime, becomes linearly disjoint over $F$ after removing an explicit finite subfamily of fields.然后，我们为这个有限的亚家族提供了必要的条件，以纠缠于$ f $，当$ f = k $时总是满足的。在这种情况下，在进一步的假设下，椭圆曲线$ e $是从$ \ mathbb {q} $的基础变化中获得的，我们详细描述了$ e $的部门家庭中的纠缠。
For every elliptic curve $E$ which has complex multiplication (CM) and is defined over a number field $F$ containing the CM field $K$, we prove that the family of $p^{\infty}$-division fields of $E$, with $p \in \mathbb{N}$ prime, becomes linearly disjoint over $F$ after removing an explicit finite subfamily of fields. We then give a necessary condition for this finite subfamily to be entangled over $F$, which is always met when $F = K$. In this case, and under the further assumption that the elliptic curve $E$ is obtained as a base-change from $\mathbb{Q}$, we describe in detail the entanglement in the family of division fields of $E$.
