Unlikely intersections on the p-adic formal ball.

We investigate generalizations along the lines of the Mordell-Lang conjecture of the author's p-adic formal Manin-Mumford results for n-dimensional p-divisible formal groups $F$ . In particular, given a finitely generated subgroup $\Gamma$ of $F\left({\overline{Q}}_p\right)$ and a closed subscheme $X\hookrightarrow F$ , we show under suitable assumptions that for any points $P\in X\left(C_p\right)$ satisfying $nP\in \Gamma$ for some $n\in N$ , the minimal such orders n are uniformly bounded whenever X does not contain a formal subgroup translate of positive dimension. In contrast, we then provide counter-examples to a full p-adic formal Mordell-Lang result. Finally, we outline some consequences for the study of the Zariski-density of sets of automorphic objects in p-adic deformations. Specifically, we do so in the context of the nearly ordinary p-adic families of cuspidal cohomological automorphic forms for the general linear group constructed by Hida.