AI 中文总结
本文研究奇素数$p$的$\mathbb{Z}_p$-扩张中有限层最大非分歧pro-$p$扩张伽罗瓦群的增广与Zassenhaus滤过，在$\mu$-不变量为正时，证明了生成元个数为$\delta p^n+O(1)$，关系个数为$O(p^n)$，并给出各滤过商的渐近行为。
AI 中文摘要
设$p$为奇素数，$K_\infty/K$为$\mathbb{Z}_p$-扩张。对每一有限层$K_n$，令$G_n$为其最大非分歧pro-$p$扩张的伽罗瓦群。我们研究当$n$趋于无穷时群$G_n$的增广滤过和Zassenhaus滤过。假设经典Iwasawa $\mu$-不变量为正。若$d_n$表示$G_n$的生成元最小个数，则存在整数$\delta>0$使得$d_n=\delta p^n+O(1)$，而关系的最小个数为$O(p^n)$。对每个固定的$m\geq2$，当$n\to\infty$时，我们证明第$m$个增广商与具有$d_n$个生成元的自由pro-$p$群具有相同的首项渐近。我们还获得了相应的Hilbert级数、增广滤过的指数增长率以及Zassenhaus商的渐近。
英文摘要
Let $p$ be an odd prime and let $K_\infty/K$ be a $\mathbb{Z}_p$-extension. For each finite layer $K_n$, let $G_n$ be the Galois group of its maximal unramified pro-$p$ extension. We study the augmentation and Zassenhaus filtrations of the groups $G_n$ as $n$ tends to infinity. Assume that the classical Iwasawa $μ$-invariant is positive. If $d_n$ denotes the minimal number of generators of $G_n$, then there is an integer $δ>0$ such that $d_n=δp^n+O(1)$, while the minimal number of relations is $O(p^n)$. For each fixed $m\geq2$, as $n\to\infty$, we show that the $m$-th augmentation quotient has the same leading asymptotic as that of a free pro-$p$ group on $d_n$ generators. We also obtain corresponding asymptotics for the Hilbert series, the exponential growth rate of the augmentation filtration, and the Zassenhaus quotients.
Commentsv1: 18 pages
