On the Iwasawa invariants for links and Kida's formula

Abstract

Analogues of Iwasawa invariants in the context of 3-dimensional topology have been studied by M.~Morishita and others. In this paper, following the dictionary of arithmetic topology, we formulate an analogue of Kida's formula on $\lambda$-invariants in a $p$-extension of ${\mathbb{Z}}_p$-fields for 3-manifolds. The proof is given in a parallel manner to Iwasawa's second proof, with use of $p$-adic representations of a finite group. In the course of our arguments, we introduce the notion of a branched ${\mathbb{Z}}_p$-cover as an inverse system of cyclic branched $p$-covers of 3-manifolds, generalize the Iwasawa type formula, and compute the Tate cohomology of 2-cycles explicitly.

26 pages, 1 figure; Minor modifications

Cited by in corpus (4)

• On pro-$p$ link groups of number fields
• $p$-adic Mahler measure and $\mathbb{Z}$-covers of links
• The Iwasawa invariants of ${\mathbb{Z}}_p^d$-covers of links
• The $p$-adic limits of class numbers in ${\mathbb{Z}}_p$-towers