Bounded height in pencils of finitely generated subgroups

Abstract

In this article we prove a general bounded height result for specializations in finitely generated subgroups varying in families which complements and sharpens the toric Mordell–Lang theorem by replacing finiteness with emptiness, for the intersection of varieties and subgroups, all moving in a pencil, except for bounded height values of the parameters (and excluding identical relations). More precisely, an instance of the result is as follows. Consider the torus scheme ${{\mathbb{G}}_m^r}_{/\mathcal{C}}$ over a curve $\mathcal{C}$ defined over $\overline{\mathbb{Q}}$, and let $\Gamma$ be a subgroup scheme generated by finitely many sections (satisfying some necessary conditions). Further, let $V$ be any subscheme. Then there is a bound for the height of the points $P\in \mathcal{C}\left(\overline{\mathbb{Q}}\right)$ such that, for some $\gamma \in \Gamma$ which does not generically lie in $V$, $\gamma \left(P\right)$ lies in the fiber $V_P$. We further offer some direct Diophantine applications, to illustrate once again that the results implicitly contain information absent from the previous bounds in this context.