On the properties of Northcott and Narkiewicz for elliptic curves

Abstract

In this paper, as an analog of the number field case, for an elliptic curve [Formula: see text] defined over the algebraic numbers and for any subfield [Formula: see text] of algebraic numbers, we say that [Formula: see text] has the Northcott property over [Formula: see text] if there are at most finitely many [Formula: see text]-rational points on [Formula: see text] of uniformly bounded height, and we say that [Formula: see text] has the property (P) over [Formula: see text] if for any infinite subset [Formula: see text] of [Formula: see text]-rational points on [Formula: see text], [Formula: see text] for an [Formula: see text]-endomorphism [Formula: see text] of [Formula: see text] implies that [Formula: see text] 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), but the converse is not true.