论文标题
在$ p $ - amin-mumford的猜想中
On $p$-adic versions of the Manin-Mumford Conjecture
论文作者
论文摘要
我们证明了算术几何形状的经典结果的$ p $ addic版本，表明具有浓密扭曲的亚伯式品种的不可约合的亚变量必须是按扭转点的亚组的翻译。我们在某些刚性分析空间和正式团体上分别在$ p $ - adic field $ k $或其整数$ r $的背景下这样做。特别是，我们表明，所谓的Manin-Mumford猜想推广到合适的$ p $ -ADIC分析功能的代数函数的刚度结果。在正式的环境中，这种方法使我们发现不来自阿伯利亚计划的正式团体纯粹是$ p $ - 亚种的Manin-Mumford类型结果。此外，我们观察到，泰特·沃洛奇（Tate-Voloch）猜想的版本在$ p $ ad的设置中保存：扭转点要么直接放在亚物种上，要么在$ p $ ad的距离上均匀地远离它。
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.
