On the properties of Northcott and Narkiewicz for elliptic curves

TL;DR

This paper investigates properties of elliptic curves over algebraic number fields, establishing criteria for Northcott and property (P), and demonstrating that Northcott implies property (P).

Contribution

It introduces criteria for Northcott and property (P) on elliptic curves over algebraic fields and proves that Northcott implies property (P).

Findings

Northcott property over F implies property (P).

Criteria established for both properties.

Examples illustrating the properties.

Abstract

In this paper, for an elliptic curve $E$ defined over the algebraic numbers and for any subfield $F$ of algebraic numbers, we say that $E$ has the Northcott property over $F$ if there are at most finitely many $F$-rational points on $E$ of uniformly bounded height, and we say that $E$ has the property (P) over $F$ if for any infinite subset $S$ of $F$-rational points on $E$, $f(S) = S$ for an $F$-endomorphism $f$ of $E$ implies that $f$ is an automorphism. We establish some criteria for both properties and provide typical examples. We also show that the Northcott property implies the property (P).