论文标题
图理论中基达公式的类似物
An analogue of Kida's formula in graph theory
论文作者
论文摘要
令$ \ ell $为理性的素数，让$ p：y \ rightarrow x $为有限图的盖洛伊斯盖，其galois组为有限的$ \ ell $ -group。考虑一个$ \ mathbb {z} _ {\ ell} $ - tower上方$ x $的塔，其回调沿$ p $。假设连接了回调中的所有图表，则获得了$ \ mathbb {z} _ {\ ell} $ - $ y $上方的塔。在假设塔之上$ x $ nishes的iwasawa $ $ $ $ $ invariant中，我们证明了一种公式，将$ \ m \ m arthbb {z} _ {z} _ {\ ell} $ x $ x $ x $ x $ x $ x $ x $ x $ x $ to iwasawa $λ$λ$ -Iinvariant的$ \ mathbb {z} _ {z} _ {z} _ {z} _ {z} _ {z} _ {z} _ {z} _ {z} _ {这个公式类似于古典伊瓦沙理论中基达的公式。我们基于与Cuoco的经典结果相比，对某些非交互性pro-$ \ ell $图的结构特性的研究提出了应用。我们的调查通过明确的例子说明了。
Let $\ell$ be a rational prime and let $p:Y\rightarrow X$ be a Galois cover of finite graphs whose Galois group is a finite $\ell$-group. Consider a $\mathbb{Z}_{\ell}$-tower above $X$ and its pullback along $p$. Assuming that all the graphs in the pullback are connected, one obtains a $\mathbb{Z}_{\ell}$-tower above $Y$. Under the assumption that the Iwasawa $μ$-invariant of the tower above $X$ vanishes, we prove a formula relating the Iwasawa $λ$-invariant of the $\mathbb{Z}_{\ell}$-tower above $X$ to the Iwasawa $λ$-invariant of the pullback. This formula is analogous to Kida's formula in classical Iwasawa theory. We present an application to the study of structural properties of certain noncommutative pro-$\ell$ towers of graphs, based on an analogy with classical results of Cuoco on the growth of Iwasawa invariants in $\mathbb{Z}_\ell^2$-extensions of number fields. Our investigations are illustrated by explicit examples.
