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 $\Q$

TL;DR

This paper constructs a Diophantine model of the integers within certain large subrings of number fields using points on elliptic curves, and shows that the integers are definable with only one universal quantifier.

Contribution

It demonstrates the definability of the integers in large subrings of number fields via elliptic curve indices, with a single universal quantifier, extending previous results.

Findings

Existence of a set of primes of natural density one for which indices are existentially definable.

Construction of a Diophantine model of  using elliptic curve points.

Definability of  with one universal quantifier in these subrings.

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.