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

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