Distribution of integral division points on the algebraic torus
| Autor: | Su-Ion Ih, Philipp Habegger |
|---|---|
| Rok vydání: | 2018 |
| Předmět: |
Discrete mathematics
Multiplicative group Applied Mathematics General Mathematics 010102 general mathematics Toric variety 020206 networking & telecommunications Clifford torus 02 engineering and technology Algebraic number field Complex torus 01 natural sciences Algebraic closure Algebraic cycle Algebraic torus 0202 electrical engineering electronic engineering information engineering 0101 mathematics Mathematics |
| Zdroj: | Transactions of the American Mathematical Society. 371:357-386 |
| ISSN: | 1088-6850 0002-9947 |
| Popis: | 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$. |
| Databáze: | OpenAIRE |
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