关于对数类群的μ和λ不变量
On the mu and lambda invariants of the logarithmic class group
摘要：
设$\ell$为有理数素数。假设Gross-Kuz'min猜想沿着数域$K$的$\Zl$ -扩张$K\_{\infty}$，我们证明了存在整数$\mut$，$\lat$和$\widetildeν$使得$K\_{\infty}$的$n$ -层$K\_{n}$的对数类群$\Clog{n}$的阶$\ell^{\tilde{e}\_{n}}$的指数$\tilde{e}\_{n}$由$\tilde{e}\_{n}=\widetildeμ\ell^{n}+\widetildeλ n + \widetildeν$给出，只要$n$足够大.本文给出了一类$\Zl$ -扩张的经典不变量$μ$和$λ$与它们的对数不变量$\mut$和$\lat$之间的关系。此外，我们提供了分圆和非分圆的情况下的数值例子。
英文摘要：
Let $\ell$ be a rational prime number. Assuming the Gross-Kuz'min conjecture along a $\Zl$-extension $K\_{\infty}$ of a number field $K$, we show that there exist integers $\mut$, $\lat$ and $\widetildeν$ such that the exponent $\tilde{e}\_{n}$ of the order $\ell^{\tilde{e}\_{n}}$ of the logarithmic class group $\Clog{n}$ for the $n$-th layer $K\_{n}$ of $K\_{\infty}$ is given by $\tilde{e}\_{n}=\widetildeμ\ell^{n}+\widetildeλ n + \widetildeν$, for $n$ big enough. We show some relations between the classical invariants $μ$ and $λ$, and their logarithmic counterparts $\mut$ and $\lat$ for some class of $\Zl$-extensions. Additionally, we provide numerical examples for the cyclotomic and the non-cyclotomic case.
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