Unlikely intersections on the $p$-adic formal ball

Abstract

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 $\mathcal{F}$. In particular, given a finitely generated subgroup $\Gamma$ of $\mathcal{F}({\overline{\mathbb{Q}}}_p)$ and a closed subscheme $X\hookrightarrow \mathcal{F}$, we show under suitable assumptions that for any points $P\in X({\mathbb{C}}_p)$ satisfying $nP\in \Gamma$ for some $n\in \mathbb{N}$, 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.

arXiv comment19 pages, section 2.3. and consequences have been revised to correct an erroneous result