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. [ABSTRACT FROM AUTHOR]