On the properties of Northcott and Narkiewicz for elliptic curves

In this paper, as an analog of the number field case, 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), but the converse is not true.