Unlikely intersections on the p-adic formal ball

Abstract

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