We establish $p$-adic versions of the Manin–Mumford conjecture, which states 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 or its ring of integers, respectively. In particular, we show that the underlying rigidity results for algebraic functions generalize to suitable $p$-adic analytic functions. This 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: torsion points either lie squarely on a subscheme or are uniformly bounded away from it in the $p$-adic distance.

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关于马宁-芒福德猜想的 p 进版本

我们建立了 Manin-Mumford 猜想的 $p$-adic 版本，该猜想表明具有密集扭转的阿贝尔簇的不可约子簇必须是子群通过扭转点的平移。我们分别在 $p$-adic 域或其整数环上的某些刚性分析空间和形式群的上下文中这样做。特别是，我们表明代数函数的基本刚性结果可以推广到合适的 $p$-adic 分析函数。这使我们发现了不是来自阿贝尔方案的正式群的纯 $p$-adic Manin-Mumford 型结果。此外，我们观察到 Tate-Voloch 猜想的一个版本成立：扭转点要么直接位于子方案上，要么在 $p$-adic 距离内均匀地与它有界。