Counting points of bounded relative height

Abstract

Let L/K be an extension of number fields and let e O-L/K(*) be the subgroup of the unit group O-L(*) consisting of the elements that are roots of L units of O-K(*). Denote by N(L/K, B) the number of points in P-1(L)/(O)(L/K)(*) with relative height in the sense of Berge-Martinet at most B. Here P-1(L) stands for the one-dimensional projective space over L. In this paper is proved the formula N(L/K, B) = CB2 + O(B2-1/[L:Q}), where C is a constant given in terms of invariants of L/K such as the regulators, class number and discriminant.