2020/07/04 by Serban, Vlad · 1 citation
#11S31 #13F25 (Secondary) #14L05 (Primary) 11G10 #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2007.02069
We prove p-adic versions of a classical result in arithmetic geometry stating that an irreducible subvariety of an abelian variety with dense torsion has to be the translate of a subgroup by a torsion point. We do so in the context of certain rigid analytic spaces and formal groups over a p-adic field K or its ring of integers R, respectively. In particular, we show that the rigidity results for algebraic functions underlying the so-called Manin-Mumford Conjecture generalize to suitable p-adic analytic functions. In the formal setting, this approach leads us to uncover purely p-adic Manin-Mumford type results for formal groups not coming from abelian schemes. Moreover, we observe that a version of the Tate-Voloch Conjecture holds in the p-adic setting: torsion points either lie squarely on a subscheme or are uniformly bounded away from it in the p-adic distance.