利用椭圆曲线上点的指数构造$\Z$的丢番图模型并在包括$\Q$在内的数域的极大子环中用一个全称量词定义$\Z$
Using Indices of Points on an Elliptic Curve to Construct A Diophantine Model of $\Z$ and Define $\Z$ Using One Universal Quantifier in Very Large Subrings of Number Fields, Including $\Q$
摘要：
令$K$是数域，并且令$E$是在$K$上定义的秩为1的椭圆曲线。对于$K$的素数集合$\calW_K$，令$O_{K,\calW_K}=\{x\in K: \ord_{\pp}x \geq 0, \forall \pp \not \in \calW_K\}$。设$P \in E(K)$是$E(K)$模扭子群的生成元.设$(x_n(P),y_n(P))$是$[n]P$相对于$E$的固定Weierstrass方程的仿射坐标。证明了存在自然密度为1的$K$的素数集合$\calW_K$，使得在$O_{K,\calW_K}$中指数的乘法（关于$P$的某个固定倍数）是存在可定义的，因此这些指数可以用来构造$\Z$的丢番图模型.我们还表明，$\Z$是可定义的$O_{K,\calW_K}$使用只有一个全称量词。这两种方法都可以将使用指数和$\Z$的一阶定义构造丢番图模型提升到$K$的任意无限扩展$K_{\infty}$中的$O_{K,\calW_K}$的整数闭包，只要$E(K_{\infty})$是随机生成的且秩为1.
英文摘要：
Let $K$ be a number field and let $E$ be an elliptic curve defined and of rank one over $K$. For a set $\calW_K$ of primes of $K$, let $O_{K,\calW_K}=\{x\in K: \ord_{\pp}x \geq 0, \forall \pp \not \in \calW_K\}$. Let $P \in E(K)$ be a generator of $E(K)$ modulo the torsion subgroup. Let $(x_n(P),y_n(P))$ be the affine coordinates of $[n]P$ with respect to a fixed Weierstrass equation of $E$. We show that there exists a set $\calW_K$ of primes of $K$ of natural density one such that in $O_{K,\calW_K}$ multiplication of indices (with respect to some fixed multiple of $P$) is existentially definable and therefore these indices can be used to construct a Diophantine model of $\Z$. We also show that $\Z$ is definable over $O_{K,\calW_K}$ using just one universal quantifier. Both, the construction of a Diophantine model using the indices and the first-order definition of $\Z$ can be lifted to the integral closure of $O_{K,\calW_K}$ in any infinite extension $K_{\infty}$ of $K$ as long as $E(K_{\infty})$ is finitely generated and of rank one.
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。
