论文标题
与椭圆曲线相关的序列的原始划分与复杂乘法
Primitive divisors of sequences associated to elliptic curves with complex multiplication
论文作者
论文摘要
让$ p $和$ q $是在数字字段$ k $上定义的椭圆曲线上的两个点。对于$α\ in \ text {end}（e）$，将$b_α$定义为$ \ mathcal {o} _k $ - ink $ - 集成理想，由$ x（α（p）+q）的分母生成。令$ \ mathcal {o} $为$ \ text {end}（e）$的子来$，这是一个dedekind域。我们将研究序列$ \ {B_α\} _ {α\ in \ Mathcal {O}} $。我们将证明，对于\ Mathcal {o} $有限的所有$α\，理想$b_α$在$ p $是一个非扭转点时具有原始除数，并且存在两个内态$ g \ neq 0 $和$ f $，因此该$ g \ neq 0 $和$ f $ f $ f $ f（p）= g（q）$。这是对椭圆性分序序列的先前结果的概括。
Let $P$ and $Q$ be two points on an elliptic curve defined over a number field $K$. For $α\in \text{End}(E)$, define $B_α$ to be the $\mathcal{O}_K$-integral ideal generated by the denominator of $x(α(P)+Q)$. Let $\mathcal{O}$ be a subring of $\text{End}(E)$, that is a Dedekind domain. We will study the sequence $\{B_α\}_{α\in \mathcal{O}}$. We will show that, for all but finitely many $α\in \mathcal{O}$, the ideal $B_α$ has a primitive divisor when $P$ is a non-torsion point and there exist two endomorphisms $g\neq 0$ and $f$ so that $f(P)=g(Q)$. This is a generalization of previous results on elliptic divisibility sequences.
