An Infinitesimal $p$-adic Multiplicative Manin-Mumford Conjecture
An Infinitesimal $p$-adic Multiplicative Manin-Mumford Conjecture
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无穷小 $p$-adic 乘法 Manin-Mumford 猜想
DOI：
10.5802/jtnb.1030
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发表时间：
2016
期刊：
影响因子：
--
通讯作者：
Vlad Serban
中科院分区：
文献类型：
--
作者：
Vlad Serban
我们的结果涉及以乘法单位为中心的开单位$p-进多圆盘上的解析函数，并且证明了这些函数只在有限多个单位根$(zeta_1-1，\ldots，\zeta_n-1)$的$n元组处消失，除非它们沿着形式乘法群的平移消失。对于多项式函数，这源于乘法Manin-Mumford猜想。然而，我们考虑了更广泛的解析函数类；特别地，我们建立了形式环面的刚性结果。此外，我们的方法不仅适用于形式乘法群，而且还适用于Lubin-Tate形式群，并且我们将结果推广到这种情况。
Our results concern analytic functions on the open unit $p$-adic poly-disc in $\mathbb{C}^n_p$ centered at the multiplicative unit and we prove that such functions only vanish at finitely many $n$-tuples of roots of unity $(\zeta_1-1,\ldots,\zeta_n-1)$ unless they vanish along a translate of the formal multiplicative group. For polynomial functions, this follows from the multiplicative Manin-Mumford conjecture. However we allow for a much wider class of analytic functions; in particular we establish a rigidity result for formal tori. Moreover, our methods apply to Lubin-Tate formal groups beyond just the formal multiplicative group and we extend the results to this setting.