Height Uniformity for Algebraic Points on Curves
We recall the main result of L. Caporaso, J. Harris, and B. Mazur's 1997 paper of ‘Uniformity of rational points.’ It says that the Lang conjecture on the distribution of rational points on varieties of general type implies the uniformity for the numbers of rational points on curves of genus at least 2. 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.
Key Words: ample divisor; big divisor; canonical divisor; fiber product; height; height zeta function; symmetric product; variety of general type; Vojta conjecture.