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 $\Q$

Abstract

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.