On the number of points of bounded height¶on arithmetic projective spaces

Abstract

Let K be a number field and \(\) its ring of integers. Let \(\) be a Hermitian vector bundle over \(\). In the first part of this paper we estimate the number of points of bounded height in \(\) (generalizing a result by Schanuel). We give then some applications: we estimate the number of hyperplanes and hypersurfaces of degree d>1 in \(\) of bounded height and containing a fixed linear subvariety and we estimate the number of points of height, with respect to the anticanonical line bundle, less then T (when T goes to infinity) of ℙNK blown up at a linear subspace of codimension two.