The ABC conjecture implies Vojta's height inequality for curves

[摘要] Following Elkies (Internat. Math. Res. Notices 7 (1991) 99-109) and Bombieri (Roth's theorem and the abc-conjecture, preprint, ETH Zurich, 1994), we show that the ABC conjecture implies the one-dimensional case of Vojta's height inequality. The main geometric tool is the construction of a Belyi 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 Roth's 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. (C) 2002 Elsevier Science (USA).

[发布日期] 2002-08-01

[关键词] ABC conjecture;the error term in the ABC conjecture;Vojta's height inequality;Diophantine approximation;Roth's theorem;type of an algebraic number;Mordell's conjecture;effective Mordell