Height uniformity for integral points on elliptic curves

We recall the result of D. Abramovich and its generalization by P. Pacelli on the uniformity for stably integral points on elliptic curves. It says that the Lang-Vojta conjecture on the distribution of integral points on a variety of logarithmic general type implies the uniformity for the numbers of stably integral points on elliptic curves. In this paper we will investigate its analogue for their heights under the assumption of the Vojta conjecture. Basically, we will show that the Vojta conjecture gives a naturally expected simple uniformity for their heights.

     

Points of Bounded Height on Equivariant Compactifications of Vector Groups, I

We prove asymptotic formulas for the number of rational points of bounded height on certain equivariant compactifications of the affine plane.     

The ABC Conjecture Implies Vojta's Height Inequality for Curves

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.