An arithmetic property of torsion subvarieties of an algebraic torus
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Let $K$ be a number field with an algebraic closure $\overline K$ and let $S$ be a finite set of places of $K$, including all the infinite ones. Let $N \geq 1$ be an integer and let $D$ be a nonzero effective cycle of ${\mathbb G}_{\rm m}^N (\overline K)$. If $\textup{Supp} (D)$ has at least one point whose coordinates are multiplicatively independent, for example, if $D = (\alpha)$ and $\alpha \in {\mathbb G}_{\rm m}^N (\overline K)$ does not lie on any proper torsion subvarieties (= translates of proper subtori by torsion points) of ${\mathbb G}_{\rm m}^N$, then we will show that the set $$ \begin{array}{cccccc} \bigcup_{{\scriptsize{ \hspace{-0.1cm} \begin{array}{ccc} Z \subset {\mathbb G}_{\rm m}^N \textup{ a torsion hypersurface} \\ \textup{such that } \overline Z \cap \overline {D} = \emptyset \textup{ (i.e., $\overline{D} |_{\overline Z} = 0$)} \end{array}} }} \hspace{-0.1cm} Z(\overline K) \end{array} $$ is not Zariski dense in ${\mathbb G}_{\rm m}^N$, where the bars indicate the respective Zariski closures of $Z$ and $D$ in ${\mathbb G}_{{\rm m}, \mathcal O_{K,S}}^N$, an $\mathcal O_{K,S}$-integral model of ${\mathbb G}_{\rm m}^N$ over $K$. We will also pose a conjecture, which generalizes this result, in the context of semiabelian varieties.
Abstract
Let $K$ be a number field with an algebraic closure $\overline K$ and let $S$ be a finite set of places of $K$, including all the infinite ones. Let $N \geq 1$ be an integer and let $D$ be a nonzero effective cycle of ${\mathbb G}_{\rm m}^N (\overline K)$. If $\textup{Supp} (D)$ has at least one point whose coordinates are multiplicatively independent, for example, if $D = (\alpha)$ and $\alpha \in {\mathbb G}_{\rm m}^N (\overline K)$ does not lie on any proper torsion subvarieties (= translates of proper subtori by torsion points) of ${\mathbb G}_{\rm m}^N$, then we will show that the set $$ \begin{array}{cccccc} \bigcup_{{\scriptsize{ \hspace{-0.1cm} \begin{array}{ccc} Z \subset {\mathbb G}_{\rm m}^N \textup{ a torsion hypersurface} \\ \textup{such that } \overline Z \cap \overline {D} = \emptyset \textup{ (i.e., $\overline{D} |_{\overline Z} = 0$)} \end{array}} }} \hspace{-0.1cm} Z(\overline K) \end{array} $$ is not Zariski dense in ${\mathbb G}_{\rm m}^N$, where the bars indicate the respective Zariski closures of $Z$ and $D$ in ${\mathbb G}_{{\rm m}, \mathcal O_{K,S}}^N$, an $\mathcal O_{K,S}$-integral model of ${\mathbb G}_{\rm m}^N$ over $K$. We will also pose a conjecture, which generalizes this result, in the context of semiabelian varieties.