On the Iwasawa invariants for links and Kida's formula

TL;DR

This paper develops an analogue of Kida's formula for Iwasawa invariants in the setting of 3-manifolds, using $p$-adic representations and branched $bZ_p$-covers, bridging number theory and topology.

Contribution

It formulates a topological analogue of Kida's formula for $bZ_p$-extensions of 3-manifolds, introducing branched $bZ_p$-covers and computing related cohomology.

Findings

Established a topological analogue of Kida's formula for $bZ_p$-extensions.

Introduced the notion of branched $bZ_p$-covers for 3-manifolds.

Computed Tate cohomology of 2-cycles explicitly.

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.