A height inequality for rational points on elliptic curves implied by   the abc-conjecture
  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. 
Ulf K\"uhn, J. Steffen M\"uller
math.NT