Distribution of integral division points on the algebraic torus

Abstract

Let $ K$ be a number field with algebraic closure $ \overline K$, and let $ S$ be a finite set of places of $ K$ containing all the infinite ones. Let $ {\it\Gamma }_0$ be a finitely generated subgroup of $ {\mathbb{G}}_{\textup {m}} (\overline K)$, and let $ {\it\Gamma } \subset {\mathbb{G}}_{\textup {m}} (\overline K)$ be the division group attached to $ {\it\Gamma }_0$. Here is an illustration of what we will prove in this article. Fix a proper closed subinterval $ I$ of $ [0, \infty )$ and a nonzero effective divisor $ D$ on $ {\mathbb{G}}_{\textup {m}}$ which is not the translate of any torsion divisor on the algebraic torus $ {\mathbb{G}}_{\textup {m}}$ by any point of $ {\it\Gamma }$ with height belonging to $ I$. Then we prove a statement which easily implies that the set of ``integral division points on $ {\mathbb{G}}_{\textup {m}}$ with height near $ I$'', i.e., the set of points of $ {\it\Gamma }$ with (standard absolute logarithmic Weil) height in $ J$ which are $ S$-integral on $ {\mathbb{G}}_{\textup {m}}$ relative to $ D,$ is finite for some fixed subinterval $ J$ of $ [0, \infty )$ 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 $ {\mathbb{G}}_{\textup {m}}$. Finally, we also propose an analogous version for a dynamical system on $ {\mathbb{P}}^1$.