Borne de hauteur semi-effective pour le problème de Mordell-Lang dans un tore

Abstract

If $X$ is a closed subscheme of torus $\mathbb{G}_{m,K}^g$ on a number field $K$, we define $Z_X$ as the union of all translated positive dimensional subtori contained in $X$. Vojta proved that the set of integral points on $X\setminus Z_X$ is finite. In this article, we consider a family of closed subschemes $V\to P$ of $\mathbb{G}_{m,K}^g$. We prove a semi-effective bound for the height of the integer points in $V_p\setminus Z_{V_p}$. We prove this result in the more general context of the Mordell-Lang plus Bogomolov problem and we obtain a corollary on the unit equation.