✦ LIBER ✦
The ABC Conjecture Implies Vojta's Height Inequality for Curves
✍ Scribed by Machiel Van Frankenhuysen
- Publisher
- Elsevier Science
- Year
- 2002
- Tongue
- English
- Weight
- 158 KB
- Volume
- 95
- Category
- Article
- ISSN
- 0022-314X
No coin nor oath required. For personal study only.
✦ Synopsis
Following Elkies (Internat. Math. Res. Notices 7 (1991) 99-109) and Bombieri (Roth's theorem and the abc-conjecture, preprint, ETH Zu¨rich, 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 Bely$ ı ı 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.