HEIGHTS ON ELLIPTIC CURVES AND THE DIOPHANTINE EQUATION x y cz

Abstract In this paper we give sharp explicit estimates for the dierence of the Weil height and the Neron Tate height on the elliptic curve v u cu We then apply this in the proof of the fact that if c is a fourth power free integer and the rank of v u cu is then the equation x y cz has no nonzero solutions in integers Heights on the Elliptic Curve v u cu In this section E is the elliptic curve v u cu where c is a fourth power free integer For a subgroup E Q dened below we give an absolute independent of c bound for the dierence between the Weil height and the N eron Tate height Let be I E Q where I and E Q are dened as follows Over R the elliptic curve E consists of two components and E