# Distribution of integral division points on the algebraic torus

Habegger, Philipp and Ih, Su-ion. (2019) Distribution of integral division points on the algebraic torus. Transactions of the American Mathematical Society, 371. pp. 357-386.

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Official URL: https://edoc.unibas.ch/59215/

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 ${\mathbb{P}}^1$.