p-adic unlikely intersection results and consequences

The Mordell-Lang conjecture (a theorem) is an "unlikely intersection" result stating that an irred. subvariety of a semi-abelian variety G that has dense intersection with the divisible hull of a finitely generated subgroup of G must in fact be the translate of a subgroup variety of G. We present certain p-adic incarnations of this result, chiefly in the context of formal groups Gˆ. Moreover, we outline some consequences to density questions arising via p-adic variational techniques within the Langlands programme.