DISTRIBUTION OF INTEGRAL DIVISION POINTS ON THE ALGEBRAIC TORUS

Let K be a number field with algebraic closure (K) over bar, and let S be a finite set of places of K containing all the infinite ones. Let Gamma(0) be a finitely generated subgroup of G(m)((K) over bar), and let Gamma subset of G(m)((K) over bar) be the division group attached to Gamma(0). Here is an illustration of what we will prove in this article. Fix a proper closed subinterval I of [0,infinity) and a nonzero effective divisor D on G(m) which is not the translate of any torsion divisor on the algebraic torus G(m) by any point of Gamma with height belonging to I. Then we prove a statement which easily implies that the set of "integral division points on G(m) with height near I", i.e., the set of points of Gamma with (standard absolute logarithmic Weil) height in J which are S-integral on G(m) relative to D, is finite for some fixed subinterval J of [0,infinity) properly containing I. We propose a conjecture on the nondensity of integral division points on semi-abelian varieties with prescribed height values, which generalizes some previously known conjectures as well as this finiteness result for G(m). Finally, we also propose an analogous version for a dynamical system on P-1.