On the $p$-adic limit of class numbers along a pro-$p$-extension
On the $p$-adic limit of class numbers along a pro-$p$-extension
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关于沿 pro-$p$-扩展的类数的 $p$-adic 限制
DOI：
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发表时间：
2023
期刊：
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通讯作者：
M. Ozaki
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作者：
M. Ozaki
设$K/k$是有限生成Galois群的数域$k$上的PRO-$p$-扩张，$k_0\subseteq k_1\subseteq\cdots\subseteq k_n\subseteq\cdots$是$K/k$的中间域的升序，使得$k_n/k$是正规的，$[k_n：k]<；\inty$和$\Bigcup_{n\ge 0}k_n=K$.我们将利用有限群的表示理论证明：$k_n$的类数的非$p$部分$h_n(p‘)$收敛于$p$--通常为$n\right tarrow\inty$，并且极限与$k_n$’S的选择无关。此外，当$K/k$是交换数域$k$上的割圆$\mathbb{Z}_p$-扩张时，我们将采用解析的方法，得到沿$K/k$的各种算术不变量的$p$-进给极限之间的某些神秘关系讨论了整数环的代数$K_2$-群的阶数，以及$p$-进制子与判别式的平方根之比。
Let $K/k$ be a pro-$p$-extension over a number field $k$ whose Galois group is finitely generated and $k_0\subseteq k_1\subseteq\cdots\subseteq k_n\subseteq\cdots$ an ascending sequence of intermediate fields of $K/k$ such that $k_n/k$ is normal, $[k_n:k]<\infty$ and $\bigcup_{n\ge 0} k_n=K$. We will show by using representation theory of finite groups that the non-$p$-part $h_n(p')$ of the class number of $k_n$ converges $p$-adically as $n\rightarrow\infty$, and the limit is independent to the choice of $k_n$'s. Also, in the case where $K/k$ is the cyclotomic $\mathbb{Z}_p$-extension over an abelian number field $k$, we will take an analytic approach and obtain certain enigmatic relationships between the $p$-adic limits of vaious arithmetic invariants along $K/k$, namely, the class number, the ratio of $p$-adic regulator and the square root of the discriminant, and the order of the algebraic $K_2$-group of the ring of integers.