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 $\Gamma _0$ be a finitely generated subgroup of $\mathbb {G}_{\mathrm {m}} (\overline K)$, and let $\Gamma \subset \mathbb {G}_{\mathrm {m}} (\overline K)$ 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 $[0, \infty )$ and a nonzero effective divisor $D$ on $\mathbb {G}_{\mathrm {m}}$ which is not the translate of any torsion divisor on the algebraic torus $\mathbb {G}_{\mathrm {m}}$ 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 $\mathbb {G}_{\mathrm {m}}$ 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 $\mathbb {G}_{\mathrm {m}}$ 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 $\mathbb {G}_{\mathrm {m}}$. Finally, we also propose an analogous version for a dynamical system on $\mathbb {P}^1$.