The ABC Conjecture Implies Vojtaʹs Height Inequality for Curves Original Research Article

Following Elkies (Internat. Math. Res. Notices7 (1991) 99–109) and Bombieri (Rothʹs theorem and the abc-conjecture, preprint, ETH Zürich, 1994), we show that the ABC conjecture implies the one-dimensional case of Vojtaʹs height inequality. The main geometric tool is the construction of a Belyimage function. We take care to make explicit the effectivity of the result: we show that an effective version of the ABC conjecture would imply an effective version of Rothʹs theorem, as well as giving an (in principle) explicit bound on the height of rational points on an algebraic curve of genus at least two.