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On the mu and lambda invariants of the logarithmic class group
Journal of Number Theory ( IF 0.6 ) Pub Date : 2020-11-01 , DOI: 10.1016/j.jnt.2020.02.014
José-Ibrahim Villanueva-Gutiérrez

Abstract Let l be a rational prime number. Assuming the Gross-Kuz'min conjecture along a Z l -extension K ∞ of a number field K, we show that there exist integers μ ˜ , λ ˜ and ν ˜ such that the exponent e ˜ n of the order l e ˜ n of the logarithmic class group C l ˜ n for the n-th layer K n of K ∞ is given by e ˜ n = μ ˜ l n + λ ˜ n + ν ˜ , for n big enough. We show some relations between the classical invariants μ and λ, and their logarithmic counterparts μ ˜ and λ ˜ for some class of Z l -extensions. Additionally, we provide numerical examples for the cyclotomic and the non-cyclotomic case.

中文翻译:

关于对数类群的 mu 和 lambda 不变量

摘要 令 l 为有理素数。假设沿数域 K 的 Z l 扩展 K ∞ 的 Gross-Kuz'min 猜想,我们证明存在整数 μ ˜ , λ ˜ 和 ν ˜ 使得 le ˜ n 阶数的指数 e ˜ n K ∞ 的第 n 层 K n 的对数类群 C l ∼ n 由 e ˜ n = μ ∼ ln + λ ˜ n + ν ˜ 给出,因为 n 足够大。我们展示了经典不变量 μ 和 λ 之间的一些关系,以及它们的对数对应项 μ ~ 和 λ ~ 对于某些类 Z l 扩展。此外,我们提供了分圆和非分圆情况的数值例子。
更新日期:2020-11-01
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