1 - Heights

Summary

Introduction

This chapter contains preliminary material on absolute values and the elementary theory of heights on projective varieties. Most of this material is quite standard, although we have included some of the finer results on classical heights which are not usually treated in other texts.

In Section 1.2 we start with absolute values, and places are introduced as equivalence classes of absolute values. The definitions of residue degree and ramification index are given, as well as their basic properties and behaviour with respect to finite degree extensions. In Sections 1.3 and 1.4 we introduce normalized absolute values and the all-important product formula in number fields and function fields. Section 1.5 contains the definition of the absolute Weil height in projective spaces, the characterization of points with height 0, and a general form of Liouville's inequality in diophantine approximation. Section 1.6 studies the height of polynomials and Mahler's measure and proves Gauss's lemma and its counterpart at infinity, Gelfond's lemma. Section 1.7, which can be omitted in a first reading, elaborates further on various comparison results about heights and norms of polynomials, including an interesting result of Per Enflo on ℓ₁-norms.

The presentation of the material in this chapter is self contained with the exception of Section 1.2, where the basic facts about absolute values are quoted from standard reference books (N. Bourbaki [47], Ch. VI, S. Lang [173], Ch.XII, and N. Jacobson [157], Ch.IX).