Heights on subvarieties of G ⁿ_m

Abstract

In this chapter we shall deal with algebraic subvarieties χ of G ⁿ_m and heights of algebraic points therein, without any restriction on their degree. Starting with an old problem of Lang on torsion points, we shall consider the distribution of points in χ (ℚ̅) with ‘small height’: although algebraic points in G ⁿ_m can have arbitrarily small height, it turns out that the restriction of lying in χ forces the height to be bounded below by a number c(χ) > 0, provided however we stay out of a certain exceptional Zariski-closed set in χ. This is the content of a theorem by Shou-Wu Zhang, actually analogue to a former conjecture by Bogomolov in the context of abelian varieties; it may be read as predicting a ‘discrete’ distribution of algebraic points on varieties. The alluded ‘exceptional’ sub-varieties of χ are finite unions of translates (by torsion points) of connected algebraic subgroups of G ⁿ_m , themselves isomorphic (in the algebraic group sense) to powers of G _m. We shall present an elementary proof of this theorem, studying along the way the simple theory of algebraic subgroups of G ⁿ_m . We shall also sketch a different approach to Zhang’s theorem, through equidistribution of the Galois conjugates of points of small height. Finally, Zhang’s theorem will be applied to gain uniformity in the estimation of the number of solutions of small height of the S-unit equation.