Heights and Rational Points

Abstract

We wish to determine all solutions of an equation f(x, y) = 0, where f is a polynomial with integer coefficients, for instance, and the solutions are in some domain. Part of determining the solutions consists in estimating the size of such solutions, in various ways. For instance if x, y are to be elements of the ring of integers Z, then we can estimate the absolute values |x|, |y| or better the maximum max(|x|, |y|). If x, y are taken to be rational numbers, we estimate the maximum of the absolute values of the numerators and denominators of x and y, written as reduced fractions. Thus we are led to define the size in a fairly general context, involving several variables and more general domains than the integers or rational numbers. The size will be defined technically by a notion called the height. Consider projective space P n , and first deal with the rational numbers, so we consider P n (Q). Let P∈P n Q), and let (x 0,...;,x n ) denote projective coordinates of P. Then these projective coordinates X j can be selected to be integers, and after dividing out by the greatest common divisor, we may assume that (x 0 ,...;,x n ) are relatively prime integers. In other words, (x 0 ,...;,x n have no prime factor in common. Then we define the height (or logarithmic height) For example, take n = 1. A point in P1(Q) — {∞} can be represented by coordinates (1,x) where x is a rational number. Write x = x 0 /x 1 where x 0 , x 1 are relatively prime integers. Then The choice of relatively prime integers to represent a point in projective space works well over the rational numbers, but does not work in more general fields, so we have to describe the height in another way, in terms of absolute values other than the ordinary absolute value. We do this fairly generally.