A height inequality for rational points on elliptic curves implied by the abc-conjecture

Abstract

In this short note we show that the uniform $abc$-conjecture puts strong restrictions on the coordinates of rational points on elliptic curves. For the proof we use a variant of Vojta's height inequality formulated by Mochizuki. As an application, we generalize a result of Silverman on elliptic non-Wieferich primes.