论文标题
Iwasawa不变的渐近生长在数字字段的非交通塔中
Asymptotic growth of Iwasawa invariants in Noncommutative towers of number fields
论文作者
论文摘要
让$ p $是一个奇怪的素数，$ f $是一个数字字段，并考虑一个统一的无限pro- $ p $扩展名$ f_ \ infty $ f $ $ f $，galois group $ g = gal（f_ \ infty/f）$。令\ [g = g_0 \ supset g_1 \ supset \ dots \ supset g_n \ supset g_ {n+1} \ supset \ dots \ dots \ dots \ dots \ dots \ dots \ dots \]是$ p $ p $ central系列的$ g $和set $ f_n：$ f_n：= f_ \ infty^f_ \ infty^{g_n} $。假设$ g $是统一的，而$ f_ \ infty $包含$ f $的cyclotomic $ \ mathbb {z} _p $ - extension。用$ a_n $表示$ p $ - $ \ mathbb {z} _p $ - f_n $的extension的类组的主要部分。 $ f_n $的$λ$ -INVARIANT与$ a_n $的corank作为$ \ Mathbb {z} _p $ -module。假设Cyclotomic $ \ Mathbb {z} _p $ - extension $ f $等于$ 0 $的iWasawa $ $ $ $ invariant。然后，$ f_n $的$ $ $ invariant的$ \ mathbb {z} _p $ - extension也是$ 0 $，而$ a_n $ is osomorphic is osomorphic to $ \ weft（\ MathBb {q} _p/\ Mathbb {z} _ Mathbb {z} _ {z} $ pright）我们研究$λ_n$的渐近增长为$ n $ to $ \ infty $。
Let $p$ be an odd prime, $F$ be a number field and consider a uniform infinite pro-$p$ extension $F_\infty$ of $F$ with Galois group $G=Gal(F_\infty/F)$. Let \[G=G_0\supset G_1\supset\dots \supset G_n\supset G_{n+1}\supset \dots\] be the descending $p$ central series of $G$ and set $F_n:=F_\infty^{G_n}$. Assume that $G$ is uniform and that $F_\infty$ contains the cyclotomic $\mathbb{Z}_p$-extension of $F$. Denote by $A_n$ the $p$-primary part of the class group of the cyclotomic $\mathbb{Z}_p$-extension of $F_n$. The $λ$-invariant of $F_n$ coincides with the corank of $A_n$ as a $\mathbb{Z}_p$-module. Assume that the Iwasawa $μ$-invariant of the cyclotomic $\mathbb{Z}_p$-extension of $F$ equal to $0$. Then, the $μ$-invariant of the cyclotomic $\mathbb{Z}_p$-extension of $F_n$ is $0$ as well and $A_n$ is isomorphic to $\left(\mathbb{Q}_p/\mathbb{Z}_p\right)^{λ_n}$. We study the asymptotic growth of $λ_n$ as $n$ goes to $\infty$.
