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 Γ of F ( Q ¯ p ) and a closed subscheme X ↪ F , we show under suitable assumptions that for any points P ∈ X ( C p ) satisfying n P ∈ Γ for some n ∈ 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.