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 Xhookrightarrow mathcal {F}, we show under suitable assumptions that for any points Pin X(mathbb {C}_p) satisfying nPin Gamma for some nin 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.