论文标题
$ p $ -ADIC的班级限制$ \ MATHBB {z} _p $ - tOWers
The $p$-adic limits of class numbers in $\mathbb{Z}_p$-towers
论文作者
论文摘要
本文以算术拓扑的精神讨论了韦伯的班级编号问题的变体，以连接Sinnott-Kisilevsky和Kionke的结果。令$ p $为主要数字。我们首先证明了$ p $ -adic的类数量的$ \ mathbb {z} _p $ - 全局字段的扩展以及$ \ mathbb {z} _p $ - 紧凑型3个manifold的类似结果。其次，我们使用订单统一至$ p $，$ p $ -adic -adic Googarithm和iwasawa forfrants的根源，建立了$ p $ p $ - 亚法限制的明确公式。最后，我们对圆环结，扭结和椭圆形曲线进行了彻底的研究。我们填写$ p $ -ADIC限制为$ \ Mathbb {z} $的案例列表，并找到案例，使得基本$ p $ - 类数字很小，并且$ν$''是任意大的。
This article discusses variants of Weber's class number problem in the spirit of arithmetic topology to connect the results of Sinnott--Kisilevsky and Kionke. Let $p$ be a prime number. We first prove the $p$-adic convergence of class numbers in a $\mathbb{Z}_p$-extension of a global field and a similar result in a $\mathbb{Z}_p$-cover of a compact 3-manifold. Secondly, we establish an explicit formula for the $p$-adic limit of the $p$-power-th cyclic resultants of a polynomial using roots of unity of orders prime to $p$, the $p$-adic logarithm, and the Iwasawa invariants. Finally, we give thorough investigations of torus knots, twist knots, and elliptic curves; we complete the list of the cases with $p$-adic limits being in $\mathbb{Z}$ and find the cases such that the base $p$-class numbers are small and $ν$'s are arbitrarily large.
