On $p$-adic versions of the Manin-Mumford Conjecture

TL;DR

This paper establishes $p$-adic analogues of the Manin-Mumford Conjecture within rigid analytic spaces and formal groups, revealing new $p$-adic phenomena and extending classical results to a non-Archimedean context.

Contribution

It introduces $p$-adic versions of the Manin-Mumford Conjecture for rigid analytic spaces and formal groups, including cases not arising from abelian schemes.

Findings

Proves $p$-adic Manin-Mumford type results for formal groups.

Shows a $p$-adic version of the Tate-Voloch Conjecture.

Generalizes rigidity results for algebraic functions to $p$-adic analytic functions.

Abstract

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.