The ABC Conjecture Implies Vojta's Height Inequality for Curves |
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Authors: | Machiel Van Frankenhuysen |
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Institution: | Department of Mathematics, Rutgers University, 110 Frelinghuysen Road, Piscataway, New Jersey, 08854-8019, f1machiel@math.rutgers.eduf1 |
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Abstract: | Following Elkies (Internat. Math. Res. Notices7 (1991) 99-109) and Bombieri (Roth's theorem and the abc-conjecture, preprint, ETH Zürich, 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 Belyǐ 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. |
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Keywords: | 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 |
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